diff --git a/doc/glossary.rst b/doc/glossary.rst
index 010f16a361531..2b4c6af0d1866 100644
--- a/doc/glossary.rst
+++ b/doc/glossary.rst
@@ -644,9 +644,8 @@ General Concepts
 
         Note that for most distance metrics, we rely on implementations from
         :mod:`scipy.spatial.distance`, but may reimplement for efficiency in
-        our context.  The :mod:`neighbors` module also duplicates some metric
-        implementations for integration with efficient binary tree search data
-        structures.
+        our context. The :class:`metrics.DistanceMetric` interface is used to implement
+        distance metrics for integration with efficient neighbors search.
 
     pd
         A shorthand for `Pandas <https://pandas.pydata.org>`_ due to the
@@ -1023,7 +1022,7 @@ such as:
 
 Further examples:
 
-* :class:`neighbors.DistanceMetric`
+* :class:`metrics.DistanceMetric`
 * :class:`gaussian_process.kernels.Kernel`
 * ``tree.Criterion``
 
diff --git a/doc/modules/classes.rst b/doc/modules/classes.rst
index 72b67b23e8dc3..b7000bcf7cbb2 100644
--- a/doc/modules/classes.rst
+++ b/doc/modules/classes.rst
@@ -1058,6 +1058,16 @@ further details.
 
    metrics.consensus_score
 
+Distance metrics
+----------------
+
+.. currentmodule:: sklearn
+
+.. autosummary::
+   :toctree: generated/
+   :template: class.rst
+
+   metrics.DistanceMetric
 
 Pairwise metrics
 ----------------
@@ -1317,7 +1327,6 @@ Model validation
    :template: class.rst
 
    neighbors.BallTree
-   neighbors.DistanceMetric
    neighbors.KDTree
    neighbors.KernelDensity
    neighbors.KNeighborsClassifier
diff --git a/doc/modules/density.rst b/doc/modules/density.rst
index 115d318183577..9e542b803ef68 100644
--- a/doc/modules/density.rst
+++ b/doc/modules/density.rst
@@ -136,9 +136,9 @@ The form of these kernels is as follows:
   :math:`K(x; h) \propto \cos(\frac{\pi x}{2h})` if :math:`x < h`
 
 The kernel density estimator can be used with any of the valid distance
-metrics (see :class:`~sklearn.neighbors.DistanceMetric` for a list of available metrics), though
-the results are properly normalized only for the Euclidean metric.  One
-particularly useful metric is the
+metrics (see :class:`~sklearn.metrics.DistanceMetric` for a list of
+available metrics), though the results are properly normalized only
+for the Euclidean metric.  One particularly useful metric is the
 `Haversine distance <https://en.wikipedia.org/wiki/Haversine_formula>`_
 which measures the angular distance between points on a sphere.  Here
 is an example of using a kernel density estimate for a visualization
diff --git a/doc/whats_new/v1.1.rst b/doc/whats_new/v1.1.rst
index 93300bf67d4a5..1a9f773ce08df 100644
--- a/doc/whats_new/v1.1.rst
+++ b/doc/whats_new/v1.1.rst
@@ -60,6 +60,15 @@ Changelog
   message when the solver does not support sparse matrices with int64 indices.
   :pr:`21093` by `Tom Dupre la Tour`_.
 
+:mod:`sklearn.metrics`
+......................
+
+- |API| :class:`metrics.DistanceMetric` has been moved from
+  :mod:`sklearn.neighbors` to :mod:`sklearn.metric`.
+  Using `neighbors.DistanceMetric` for imports is still valid for
+  backward compatibility, but this alias will be removed in 1.3.
+  :pr:`21177` by :user:`Julien Jerphanion <jjerphan>`.
+
 :mod:`sklearn.model_selection`
 ..............................
 
diff --git a/sklearn/cluster/_agglomerative.py b/sklearn/cluster/_agglomerative.py
index 6606f370b81eb..70b3a5028169b 100644
--- a/sklearn/cluster/_agglomerative.py
+++ b/sklearn/cluster/_agglomerative.py
@@ -16,8 +16,8 @@
 
 from ..base import BaseEstimator, ClusterMixin
 from ..metrics.pairwise import paired_distances
-from ..neighbors import DistanceMetric
-from ..neighbors._dist_metrics import METRIC_MAPPING
+from ..metrics import DistanceMetric
+from ..metrics._dist_metrics import METRIC_MAPPING
 from ..utils import check_array
 from ..utils._fast_dict import IntFloatDict
 from ..utils.fixes import _astype_copy_false
diff --git a/sklearn/cluster/_hierarchical_fast.pyx b/sklearn/cluster/_hierarchical_fast.pyx
index 2a58757ce327d..11ea3294c086a 100644
--- a/sklearn/cluster/_hierarchical_fast.pyx
+++ b/sklearn/cluster/_hierarchical_fast.pyx
@@ -13,7 +13,7 @@ ctypedef np.int8_t INT8
 
 np.import_array()
 
-from ..neighbors._dist_metrics cimport DistanceMetric
+from ..metrics._dist_metrics cimport DistanceMetric
 from ..utils._fast_dict cimport IntFloatDict
 
 # C++
@@ -236,8 +236,8 @@ def max_merge(IntFloatDict a, IntFloatDict b,
 def average_merge(IntFloatDict a, IntFloatDict b,
               np.ndarray[ITYPE_t, ndim=1] mask,
               ITYPE_t n_a, ITYPE_t n_b):
-    """Merge two IntFloatDicts with the average strategy: when the 
-    same key is present in the two dicts, the weighted average of the two 
+    """Merge two IntFloatDicts with the average strategy: when the
+    same key is present in the two dicts, the weighted average of the two
     values is used.
 
     Parameters
@@ -290,13 +290,13 @@ def average_merge(IntFloatDict a, IntFloatDict b,
 
 
 ###############################################################################
-# An edge object for fast comparisons 
+# An edge object for fast comparisons
 
 cdef class WeightedEdge:
     cdef public ITYPE_t a
     cdef public ITYPE_t b
     cdef public DTYPE_t weight
-    
+
     def __init__(self, DTYPE_t weight, ITYPE_t a, ITYPE_t b):
         self.weight = weight
         self.a = a
@@ -326,7 +326,7 @@ cdef class WeightedEdge:
             return self.weight > other.weight
         elif op == 5:
             return self.weight >= other.weight
-        
+
     def __repr__(self):
         return "%s(weight=%f, a=%i, b=%i)" % (self.__class__.__name__,
                                               self.weight,
@@ -475,7 +475,7 @@ def mst_linkage_core(
 
     dist_metric: DistanceMetric
         A DistanceMetric object conforming to the API from
-        ``sklearn.neighbors._dist_metrics.pxd`` that will be
+        ``sklearn.metrics._dist_metrics.pxd`` that will be
         used to compute distances.
 
     Returns
@@ -534,4 +534,3 @@ def mst_linkage_core(
         current_node = new_node
 
     return np.array(result)
-
diff --git a/sklearn/cluster/tests/test_hierarchical.py b/sklearn/cluster/tests/test_hierarchical.py
index 92f92dc3736e3..3525643383c26 100644
--- a/sklearn/cluster/tests/test_hierarchical.py
+++ b/sklearn/cluster/tests/test_hierarchical.py
@@ -17,7 +17,7 @@
 from scipy.sparse.csgraph import connected_components
 
 from sklearn.metrics.cluster import adjusted_rand_score
-from sklearn.neighbors.tests.test_dist_metrics import METRICS_DEFAULT_PARAMS
+from sklearn.metrics.tests.test_dist_metrics import METRICS_DEFAULT_PARAMS
 from sklearn.utils._testing import assert_almost_equal, create_memmap_backed_data
 from sklearn.utils._testing import assert_array_almost_equal
 from sklearn.utils._testing import ignore_warnings
@@ -31,6 +31,7 @@
     _fix_connectivity,
 )
 from sklearn.feature_extraction.image import grid_to_graph
+from sklearn.metrics import DistanceMetric
 from sklearn.metrics.pairwise import (
     PAIRED_DISTANCES,
     cosine_distances,
@@ -38,7 +39,7 @@
     pairwise_distances,
 )
 from sklearn.metrics.cluster import normalized_mutual_info_score
-from sklearn.neighbors import kneighbors_graph, DistanceMetric
+from sklearn.neighbors import kneighbors_graph
 from sklearn.cluster._hierarchical_fast import (
     average_merge,
     max_merge,
diff --git a/sklearn/metrics/__init__.py b/sklearn/metrics/__init__.py
index 46958ea4ef7f8..e4339229c5b64 100644
--- a/sklearn/metrics/__init__.py
+++ b/sklearn/metrics/__init__.py
@@ -36,6 +36,8 @@
 from ._classification import brier_score_loss
 from ._classification import multilabel_confusion_matrix
 
+from ._dist_metrics import DistanceMetric
+
 from . import cluster
 from .cluster import adjusted_mutual_info_score
 from .cluster import adjusted_rand_score
@@ -115,6 +117,7 @@
     "davies_bouldin_score",
     "DetCurveDisplay",
     "det_curve",
+    "DistanceMetric",
     "euclidean_distances",
     "explained_variance_score",
     "f1_score",
diff --git a/sklearn/neighbors/_dist_metrics.pxd b/sklearn/metrics/_dist_metrics.pxd
similarity index 87%
rename from sklearn/neighbors/_dist_metrics.pxd
rename to sklearn/metrics/_dist_metrics.pxd
index 5b223f8c6d8a8..61bb4fb2fe011 100644
--- a/sklearn/neighbors/_dist_metrics.pxd
+++ b/sklearn/metrics/_dist_metrics.pxd
@@ -1,14 +1,12 @@
-#!python
-#cython: boundscheck=False
-#cython: wraparound=False
-#cython: cdivision=True
+# cython: boundscheck=False
+# cython: cdivision=True
+# cython: initializedcheck=False
+# cython: wraparound=False
 
-cimport cython
 cimport numpy as np
-from libc.math cimport fabs, sqrt, exp, cos, pow
+from libc.math cimport sqrt, exp
 
-from ._typedefs cimport DTYPE_t, ITYPE_t, DITYPE_t
-from ._typedefs import DTYPE, ITYPE
+from ..utils._typedefs cimport DTYPE_t, ITYPE_t
 
 ######################################################################
 # Inline distance functions
@@ -60,7 +58,7 @@ cdef class DistanceMetric:
     cdef DTYPE_t dist(self, const DTYPE_t* x1, const DTYPE_t* x2,
                       ITYPE_t size) nogil except -1
 
-    cdef DTYPE_t rdist(self, DTYPE_t* x1, DTYPE_t* x2,
+    cdef DTYPE_t rdist(self, const DTYPE_t* x1, const DTYPE_t* x2,
                        ITYPE_t size) nogil except -1
 
     cdef int pdist(self, const DTYPE_t[:, ::1] X, DTYPE_t[:, ::1] D) except -1
diff --git a/sklearn/neighbors/_dist_metrics.pyx b/sklearn/metrics/_dist_metrics.pyx
similarity index 96%
rename from sklearn/neighbors/_dist_metrics.pyx
rename to sklearn/metrics/_dist_metrics.pyx
index db93263ee8eda..a8fb4c45ddd0c 100644
--- a/sklearn/neighbors/_dist_metrics.pyx
+++ b/sklearn/metrics/_dist_metrics.pyx
@@ -1,8 +1,7 @@
-#!python
-#cython: boundscheck=False
-#cython: wraparound=False
-#cython: initializedcheck=False
-#cython: cdivision=True
+# cython: boundscheck=False
+# cython: cdivision=True
+# cython: initializedcheck=False
+# cython: wraparound=False
 
 # By Jake Vanderplas (2013) <jakevdp@cs.washington.edu>
 # written for the scikit-learn project
@@ -19,7 +18,7 @@ cdef extern from "arrayobject.h":
                                      int typenum, void* data)
 
 
-cdef inline np.ndarray _buffer_to_ndarray(DTYPE_t* x, np.npy_intp n):
+cdef inline np.ndarray _buffer_to_ndarray(const DTYPE_t* x, np.npy_intp n):
     # Wrap a memory buffer with an ndarray. Warning: this is not robust.
     # In particular, if x is deallocated before the returned array goes
     # out of scope, this could cause memory errors.  Since there is not
@@ -33,8 +32,8 @@ cdef inline np.ndarray _buffer_to_ndarray(DTYPE_t* x, np.npy_intp n):
 from libc.math cimport fabs, sqrt, exp, pow, cos, sin, asin
 cdef DTYPE_t INF = np.inf
 
-from ._typedefs cimport DTYPE_t, ITYPE_t, DITYPE_t, DTYPECODE
-from ._typedefs import DTYPE, ITYPE
+from ..utils._typedefs cimport DTYPE_t, ITYPE_t, DITYPE_t, DTYPECODE
+from ..utils._typedefs import DTYPE, ITYPE
 
 
 ######################################################################
@@ -98,7 +97,7 @@ cdef class DistanceMetric:
 
     Examples
     --------
-    >>> from sklearn.neighbors import DistanceMetric
+    >>> from sklearn.metrics import DistanceMetric
     >>> dist = DistanceMetric.get_metric('euclidean')
     >>> X = [[0, 1, 2],
              [3, 4, 5]]
@@ -291,14 +290,13 @@ cdef class DistanceMetric:
 
     cdef DTYPE_t rdist(self, const DTYPE_t* x1, const DTYPE_t* x2,
                        ITYPE_t size) nogil except -1:
-        """Compute the reduced distance between vectors x1 and x2.
+        """Compute the rank-preserving surrogate distance between vectors x1 and x2.
 
         This can optionally be overridden in a base class.
 
-        The reduced distance is any measure that yields the same rank as the
-        distance, but is more efficient to compute.  For example, for the
-        Euclidean metric, the reduced distance is the squared-euclidean
-        distance.
+        The rank-preserving surrogate distance is any measure that yields the same
+        rank as the distance, but is more efficient to compute. For example, for the
+        Euclidean metric, the surrogate distance is the squared-euclidean distance.
         """
         return self.dist(x1, x2, size)
 
@@ -323,25 +321,24 @@ cdef class DistanceMetric:
         return 0
 
     cdef DTYPE_t _rdist_to_dist(self, DTYPE_t rdist) nogil except -1:
-        """Convert the reduced distance to the distance"""
+        """Convert the rank-preserving surrogate distance to the distance"""
         return rdist
 
     cdef DTYPE_t _dist_to_rdist(self, DTYPE_t dist) nogil except -1:
-        """Convert the distance to the reduced distance"""
+        """Convert the distance to the rank-preserving surrogate distance"""
         return dist
 
     def rdist_to_dist(self, rdist):
-        """Convert the Reduced distance to the true distance.
+        """Convert the rank-preserving surrogate distance to the distance.
 
-        The reduced distance, defined for some metrics, is a computationally
-        more efficient measure which preserves the rank of the true distance.
-        For example, in the Euclidean distance metric, the reduced distance
-        is the squared-euclidean distance.
+        The surrogate distance is any measure that yields the same rank as the
+        distance, but is more efficient to compute. For example, for the
+        Euclidean metric, the surrogate distance is the squared-euclidean distance.
 
         Parameters
         ----------
         rdist : double
-            Reduced distance.
+            Surrogate distance.
 
         Returns
         -------
@@ -351,12 +348,11 @@ cdef class DistanceMetric:
         return rdist
 
     def dist_to_rdist(self, dist):
-        """Convert the true distance to the reduced distance.
+        """Convert the true distance to the rank-preserving surrogate distance.
 
-        The reduced distance, defined for some metrics, is a computationally
-        more efficient measure which preserves the rank of the true distance.
-        For example, in the Euclidean distance metric, the reduced distance
-        is the squared-euclidean distance.
+        The surrogate distance is any measure that yields the same rank as the
+        distance, but is more efficient to compute. For example, for the
+        Euclidean metric, the surrogate distance is the squared-euclidean distance.
 
         Parameters
         ----------
@@ -366,7 +362,7 @@ cdef class DistanceMetric:
         Returns
         -------
         double
-            Reduced distance.
+            Surrogate distance.
         """
         return dist
 
@@ -519,7 +515,7 @@ cdef class ChebyshevDistance(DistanceMetric):
 
     Examples
     --------
-    >>> from sklearn.neighbors.dist_metrics import DistanceMetric
+    >>> from sklearn.metrics.dist_metrics import DistanceMetric
     >>> dist = DistanceMetric.get_metric('chebyshev')
     >>> X = [[0, 1, 2],
     ...      [3, 4, 5]]
diff --git a/sklearn/metrics/pairwise.py b/sklearn/metrics/pairwise.py
index d493ad68603ea..51cf80614cb3c 100644
--- a/sklearn/metrics/pairwise.py
+++ b/sklearn/metrics/pairwise.py
@@ -780,7 +780,7 @@ def haversine_distances(X, Y=None):
     array([[    0.        , 11099.54035582],
            [11099.54035582,     0.        ]])
     """
-    from ..neighbors import DistanceMetric
+    from ..metrics import DistanceMetric
 
     return DistanceMetric.get_metric("haversine").pairwise(X, Y)
 
diff --git a/sklearn/metrics/setup.py b/sklearn/metrics/setup.py
index df1a1caad17e0..69925a3590be6 100644
--- a/sklearn/metrics/setup.py
+++ b/sklearn/metrics/setup.py
@@ -1,4 +1,5 @@
 import os
+import numpy as np
 
 from numpy.distutils.misc_util import Configuration
 
@@ -18,6 +19,13 @@ def configuration(parent_package="", top_path=None):
         "_pairwise_fast", sources=["_pairwise_fast.pyx"], libraries=libraries
     )
 
+    config.add_extension(
+        "_dist_metrics",
+        sources=["_dist_metrics.pyx"],
+        include_dirs=[np.get_include(), os.path.join(np.get_include(), "numpy")],
+        libraries=libraries,
+    )
+
     config.add_subpackage("tests")
 
     return config
diff --git a/sklearn/neighbors/tests/test_dist_metrics.py b/sklearn/metrics/tests/test_dist_metrics.py
similarity index 95%
rename from sklearn/neighbors/tests/test_dist_metrics.py
rename to sklearn/metrics/tests/test_dist_metrics.py
index 08298f087c216..9440abba6f848 100644
--- a/sklearn/neighbors/tests/test_dist_metrics.py
+++ b/sklearn/metrics/tests/test_dist_metrics.py
@@ -7,8 +7,7 @@
 import pytest
 
 from scipy.spatial.distance import cdist
-from sklearn.neighbors import DistanceMetric
-from sklearn.neighbors import BallTree
+from sklearn.metrics import DistanceMetric
 from sklearn.utils import check_random_state
 from sklearn.utils._testing import create_memmap_backed_data
 from sklearn.utils.fixes import sp_version, parse_version
@@ -230,16 +229,6 @@ def test_pyfunc_metric():
     assert_array_almost_equal(D1_pkl, D2_pkl)
 
 
-def test_bad_pyfunc_metric():
-    def wrong_distance(x, y):
-        return "1"
-
-    X = np.ones((5, 2))
-    msg = "Custom distance function must accept two vectors"
-    with pytest.raises(TypeError, match=msg):
-        BallTree(X, metric=wrong_distance)
-
-
 def test_input_data_size():
     # Regression test for #6288
     # Previously, a metric requiring a particular input dimension would fail
diff --git a/sklearn/neighbors/__init__.py b/sklearn/neighbors/__init__.py
index 8a0934eecf142..340910008f75c 100644
--- a/sklearn/neighbors/__init__.py
+++ b/sklearn/neighbors/__init__.py
@@ -5,7 +5,7 @@
 
 from ._ball_tree import BallTree
 from ._kd_tree import KDTree
-from ._dist_metrics import DistanceMetric
+from ._distance_metric import DistanceMetric
 from ._graph import kneighbors_graph, radius_neighbors_graph
 from ._graph import KNeighborsTransformer, RadiusNeighborsTransformer
 from ._unsupervised import NearestNeighbors
diff --git a/sklearn/neighbors/_binary_tree.pxi b/sklearn/neighbors/_binary_tree.pxi
index 9f90414994550..f25da86e2148c 100644
--- a/sklearn/neighbors/_binary_tree.pxi
+++ b/sklearn/neighbors/_binary_tree.pxi
@@ -153,11 +153,16 @@ import numpy as np
 import warnings
 from ..utils import check_array
 
-from ._typedefs cimport DTYPE_t, ITYPE_t, DITYPE_t
-from ._typedefs import DTYPE, ITYPE
+from sklearn.utils._typedefs cimport DTYPE_t, ITYPE_t, DITYPE_t
+from sklearn.utils._typedefs import DTYPE, ITYPE
 
-from ._dist_metrics cimport (DistanceMetric, euclidean_dist, euclidean_rdist,
-                             euclidean_dist_to_rdist, euclidean_rdist_to_dist)
+from ..metrics._dist_metrics cimport (
+    DistanceMetric,
+    euclidean_dist,
+    euclidean_rdist,
+    euclidean_dist_to_rdist,
+    euclidean_rdist_to_dist,
+)
 
 from ._partition_nodes cimport partition_node_indices
 
@@ -878,7 +883,7 @@ def newObj(obj):
 
 ######################################################################
 # define the reverse mapping of VALID_METRICS
-from ._dist_metrics import get_valid_metric_ids
+from sklearn.metrics._dist_metrics import get_valid_metric_ids
 VALID_METRIC_IDS = get_valid_metric_ids(VALID_METRICS)
 
 
diff --git a/sklearn/neighbors/_classification.py b/sklearn/neighbors/_classification.py
index d616eaa2f32a8..056e980e63ebc 100644
--- a/sklearn/neighbors/_classification.py
+++ b/sklearn/neighbors/_classification.py
@@ -67,8 +67,8 @@ class KNeighborsClassifier(KNeighborsMixin, ClassifierMixin, NeighborsBase):
     metric : str or callable, default='minkowski'
         The distance metric to use for the tree.  The default metric is
         minkowski, and with p=2 is equivalent to the standard Euclidean
-        metric. See the documentation of :class:`DistanceMetric` for a
-        list of available metrics.
+        metric. For a list of available metrics, see the documentation of
+        :class:`~sklearn.metrics.DistanceMetric`.
         If metric is "precomputed", X is assumed to be a distance matrix and
         must be square during fit. X may be a :term:`sparse graph`,
         in which case only "nonzero" elements may be considered neighbors.
@@ -344,8 +344,8 @@ class RadiusNeighborsClassifier(RadiusNeighborsMixin, ClassifierMixin, Neighbors
     metric : str or callable, default='minkowski'
         Distance metric to use for the tree.  The default metric is
         minkowski, and with p=2 is equivalent to the standard Euclidean
-        metric. See the documentation of :class:`DistanceMetric` for a
-        list of available metrics.
+        metric. For a list of available metrics, see the documentation of
+        :class:`~sklearn.metrics.DistanceMetric`.
         If metric is "precomputed", X is assumed to be a distance matrix and
         must be square during fit. X may be a :term:`sparse graph`,
         in which case only "nonzero" elements may be considered neighbors.
diff --git a/sklearn/neighbors/_distance_metric.py b/sklearn/neighbors/_distance_metric.py
new file mode 100644
index 0000000000000..c973425d2e7b6
--- /dev/null
+++ b/sklearn/neighbors/_distance_metric.py
@@ -0,0 +1,20 @@
+# TODO: Remove this file in 1.3
+import warnings
+
+from ..metrics import DistanceMetric as _DistanceMetric
+
+
+class DistanceMetric(_DistanceMetric):
+    @classmethod
+    def _warn(cls):
+        warnings.warn(
+            "sklearn.neighbors.DistanceMetric has been moved "
+            "to sklearn.metrics.DistanceMetric in 1.0. "
+            "This import path will be removed in 1.3",
+            category=FutureWarning,
+        )
+
+    @classmethod
+    def get_metric(cls, metric, **kwargs):
+        DistanceMetric._warn()
+        return _DistanceMetric.get_metric(metric, **kwargs)
diff --git a/sklearn/neighbors/_graph.py b/sklearn/neighbors/_graph.py
index e6fdeffe3b291..9afa37b71a808 100644
--- a/sklearn/neighbors/_graph.py
+++ b/sklearn/neighbors/_graph.py
@@ -65,10 +65,11 @@ def kneighbors_graph(
         between neighbors according to the given metric.
 
     metric : str, default='minkowski'
-        The distance metric used to calculate the k-Neighbors for each sample
-        point. The DistanceMetric class gives a list of available metrics.
-        The default distance is 'euclidean' ('minkowski' metric with the p
-        param equal to 2.)
+        The distance metric to use for the tree. The default metric is
+        minkowski, and with p=2 is equivalent to the standard Euclidean
+        metric.
+        For a list of available metrics, see the documentation of
+        :class:`~sklearn.metrics.DistanceMetric`.
 
     p : int, default=2
         Power parameter for the Minkowski metric. When p = 1, this is
@@ -157,10 +158,11 @@ def radius_neighbors_graph(
         between neighbors according to the given metric.
 
     metric : str, default='minkowski'
-        The distance metric used to calculate the neighbors within a
-        given radius for each sample point. The DistanceMetric class
-        gives a list of available metrics. The default distance is
-        'euclidean' ('minkowski' metric with the param equal to 2.)
+        The distance metric to use for the tree. The default metric is
+        minkowski, and with p=2 is equivalent to the standard Euclidean
+        metric.
+        For a list of available metrics, see the documentation of
+        :class:`~sklearn.metrics.DistanceMetric`.
 
     p : int, default=2
         Power parameter for the Minkowski metric. When p = 1, this is
diff --git a/sklearn/neighbors/_partition_nodes.pxd b/sklearn/neighbors/_partition_nodes.pxd
index 522e826632824..94b02002d7a1e 100644
--- a/sklearn/neighbors/_partition_nodes.pxd
+++ b/sklearn/neighbors/_partition_nodes.pxd
@@ -1,4 +1,4 @@
-from ._typedefs cimport DTYPE_t, ITYPE_t
+from ..utils._typedefs cimport DTYPE_t, ITYPE_t
 
 cdef int partition_node_indices(
         DTYPE_t *data,
diff --git a/sklearn/neighbors/_unsupervised.py b/sklearn/neighbors/_unsupervised.py
index 6b6eec1a3112b..440ac41eb71d5 100644
--- a/sklearn/neighbors/_unsupervised.py
+++ b/sklearn/neighbors/_unsupervised.py
@@ -41,8 +41,8 @@ class NearestNeighbors(KNeighborsMixin, RadiusNeighborsMixin, NeighborsBase):
     metric : str or callable, default='minkowski'
         The distance metric to use for the tree.  The default metric is
         minkowski, and with p=2 is equivalent to the standard Euclidean
-        metric. See the documentation of :class:`DistanceMetric` for a
-        list of available metrics.
+        metric. For a list of available metrics, see the documentation of
+        :class:`~sklearn.metrics.DistanceMetric`.
         If metric is "precomputed", X is assumed to be a distance matrix and
         must be square during fit. X may be a :term:`sparse graph`,
         in which case only "nonzero" elements may be considered neighbors.
diff --git a/sklearn/neighbors/setup.py b/sklearn/neighbors/setup.py
index 85305efc29c78..aa19ba501b18d 100644
--- a/sklearn/neighbors/setup.py
+++ b/sklearn/neighbors/setup.py
@@ -32,19 +32,6 @@ def configuration(parent_package="", top_path=None):
         libraries=libraries,
     )
 
-    config.add_extension(
-        "_dist_metrics",
-        sources=["_dist_metrics.pyx"],
-        include_dirs=[numpy.get_include(), os.path.join(numpy.get_include(), "numpy")],
-        libraries=libraries,
-    )
-
-    config.add_extension(
-        "_typedefs",
-        sources=["_typedefs.pyx"],
-        include_dirs=[numpy.get_include()],
-        libraries=libraries,
-    )
     config.add_extension(
         "_quad_tree",
         sources=["_quad_tree.pyx"],
diff --git a/sklearn/neighbors/tests/test_ball_tree.py b/sklearn/neighbors/tests/test_ball_tree.py
index c751539f2a1ae..41ccff25a260e 100644
--- a/sklearn/neighbors/tests/test_ball_tree.py
+++ b/sklearn/neighbors/tests/test_ball_tree.py
@@ -4,7 +4,6 @@
 import pytest
 from numpy.testing import assert_array_almost_equal
 from sklearn.neighbors._ball_tree import BallTree
-from sklearn.neighbors import DistanceMetric
 from sklearn.utils import check_random_state
 from sklearn.utils.validation import check_array
 from sklearn.utils._testing import _convert_container
@@ -40,6 +39,8 @@
 
 
 def brute_force_neighbors(X, Y, k, metric, **kwargs):
+    from sklearn.metrics import DistanceMetric
+
     X, Y = check_array(X), check_array(Y)
     D = DistanceMetric.get_metric(metric, **kwargs).pairwise(Y, X)
     ind = np.argsort(D, axis=1)[:, :k]
diff --git a/sklearn/neighbors/tests/test_neighbors.py b/sklearn/neighbors/tests/test_neighbors.py
index a9592ff9f2c51..aa9c624cdb3cc 100644
--- a/sklearn/neighbors/tests/test_neighbors.py
+++ b/sklearn/neighbors/tests/test_neighbors.py
@@ -1823,3 +1823,15 @@ def test_pairwise_deprecated(NearestNeighbors):
     msg = r"Attribute `_pairwise` was deprecated in version 0\.24"
     with pytest.warns(FutureWarning, match=msg):
         nn._pairwise
+
+
+# TODO: Remove in 1.3
+def test_neighbors_distance_metric_deprecation():
+    from sklearn.neighbors import DistanceMetric
+    from sklearn.metrics import DistanceMetric as ActualDistanceMetric
+
+    msg = r"This import path will be removed in 1\.3"
+    with pytest.warns(FutureWarning, match=msg):
+        dist_metric = DistanceMetric.get_metric("euclidean")
+
+    assert isinstance(dist_metric, ActualDistanceMetric)
diff --git a/sklearn/neighbors/tests/test_neighbors_tree.py b/sklearn/neighbors/tests/test_neighbors_tree.py
index de34b4d230171..e043ffb730708 100644
--- a/sklearn/neighbors/tests/test_neighbors_tree.py
+++ b/sklearn/neighbors/tests/test_neighbors_tree.py
@@ -6,7 +6,7 @@
 import numpy as np
 import pytest
 
-from sklearn.neighbors import DistanceMetric
+from sklearn.metrics import DistanceMetric
 from sklearn.neighbors._ball_tree import (
     BallTree,
     kernel_norm,
diff --git a/sklearn/neighbors/_typedefs.pxd b/sklearn/utils/_typedefs.pxd
similarity index 100%
rename from sklearn/neighbors/_typedefs.pxd
rename to sklearn/utils/_typedefs.pxd
diff --git a/sklearn/neighbors/_typedefs.pyx b/sklearn/utils/_typedefs.pyx
similarity index 100%
rename from sklearn/neighbors/_typedefs.pyx
rename to sklearn/utils/_typedefs.pyx
diff --git a/sklearn/utils/setup.py b/sklearn/utils/setup.py
index c75cbe2d86495..ed78ecc5db76f 100644
--- a/sklearn/utils/setup.py
+++ b/sklearn/utils/setup.py
@@ -88,6 +88,13 @@ def configuration(parent_package="", top_path=None):
         libraries=libraries,
     )
 
+    config.add_extension(
+        "_typedefs",
+        sources=["_typedefs.pyx"],
+        include_dirs=[numpy.get_include()],
+        libraries=libraries,
+    )
+
     config.add_subpackage("tests")
 
     return config