diff --git a/doc/whats_new/v0.23.rst b/doc/whats_new/v0.23.rst
index 3de4c40e109c7..e0efe3217841d 100644
--- a/doc/whats_new/v0.23.rst
+++ b/doc/whats_new/v0.23.rst
@@ -73,6 +73,10 @@ Changelog
   `store_cv_values` is `True`.
   :pr:`15652` by :user:`Jérôme Dockès <jeromedockes>`.
 
+- |Fix| add `best_score_` attribute to :class:`linear_model.RidgeCV` and
+  :class:`linear_model.RidgeClassifierCV`.
+  :pr:`15653` by :user:`Jérôme Dockès <jeromedockes>`.
+
 :mod:`sklearn.preprocessing`
 ............................
 
diff --git a/sklearn/linear_model/_ridge.py b/sklearn/linear_model/_ridge.py
index 1c0407066048c..3b54a4eb5b3cb 100644
--- a/sklearn/linear_model/_ridge.py
+++ b/sklearn/linear_model/_ridge.py
@@ -1576,6 +1576,7 @@ def fit(self, X, y, sample_weight=None):
                                   store_cv_values=self.store_cv_values)
             estimator.fit(X, y, sample_weight=sample_weight)
             self.alpha_ = estimator.alpha_
+            self.best_score_ = estimator.best_score_
             if self.store_cv_values:
                 self.cv_values_ = estimator.cv_values_
         else:
@@ -1591,6 +1592,7 @@ def fit(self, X, y, sample_weight=None):
             gs.fit(X, y, sample_weight=sample_weight)
             estimator = gs.best_estimator_
             self.alpha_ = gs.best_estimator_.alpha
+            self.best_score_ = gs.best_score_
 
         self.coef_ = estimator.coef_
         self.intercept_ = estimator.intercept_
@@ -1693,6 +1695,9 @@ class RidgeCV(MultiOutputMixin, RegressorMixin, _BaseRidgeCV):
     alpha_ : float
         Estimated regularization parameter.
 
+    best_score_ : float
+        Mean cross-validated score of the estimator with the best alpha found.
+
     Examples
     --------
     >>> from sklearn.datasets import load_diabetes
@@ -1795,7 +1800,10 @@ class RidgeClassifierCV(LinearClassifierMixin, _BaseRidgeCV):
         ``fit_intercept = False``.
 
     alpha_ : float
-        Estimated regularization parameter
+        Estimated regularization parameter.
+
+    best_score_ : float
+        Mean cross-validated score of the estimator with the best alpha found.
 
     classes_ : array of shape (n_classes,)
         The classes labels.
diff --git a/sklearn/linear_model/tests/test_ridge.py b/sklearn/linear_model/tests/test_ridge.py
index ab45a093500df..a92e830aba66e 100644
--- a/sklearn/linear_model/tests/test_ridge.py
+++ b/sklearn/linear_model/tests/test_ridge.py
@@ -664,17 +664,31 @@ def _test_ridge_cv(filter_):
 
 @pytest.mark.parametrize(
     "ridge, make_dataset",
-    [(RidgeCV(), make_regression),
-     (RidgeClassifierCV(), make_classification)]
+    [(RidgeCV(store_cv_values=False), make_regression),
+     (RidgeClassifierCV(store_cv_values=False), make_classification)]
 )
 def test_ridge_gcv_cv_values_not_stored(ridge, make_dataset):
     # Check that `cv_values_` is not stored when store_cv_values is False
     X, y = make_dataset(n_samples=6, random_state=42)
-    ridge.set_params(store_cv_values=False)
     ridge.fit(X, y)
     assert not hasattr(ridge, "cv_values_")
 
 
+@pytest.mark.parametrize(
+    "ridge, make_dataset",
+    [(RidgeCV(), make_regression),
+     (RidgeClassifierCV(), make_classification)]
+)
+@pytest.mark.parametrize("cv", [None, 3])
+def test_ridge_best_score(ridge, make_dataset, cv):
+    # check that the best_score_ is store
+    X, y = make_dataset(n_samples=6, random_state=42)
+    ridge.set_params(store_cv_values=False, cv=cv)
+    ridge.fit(X, y)
+    assert hasattr(ridge, "best_score_")
+    assert isinstance(ridge.best_score_, float)
+
+
 def _test_ridge_diabetes(filter_):
     ridge = Ridge(fit_intercept=False)
     ridge.fit(filter_(X_diabetes), y_diabetes)