diff --git a/Lib/asyncio/runners.py b/Lib/asyncio/runners.py
index 7489a50f8e48da..ae9721e57ae17e 100644
--- a/Lib/asyncio/runners.py
+++ b/Lib/asyncio/runners.py
@@ -118,7 +118,7 @@ def run(self, coro, *, context=None):
                 events.set_event_loop(self._loop)
             return self._loop.run_until_complete(task)
         except exceptions.CancelledError:
-            if self._interrupt_count > 0 and task.uncancel() == 0:
+            if self._interrupt_count > 0 and tasks._uncancel(task) == 0:
                 raise KeyboardInterrupt()
             else:
                 raise  # CancelledError
diff --git a/Lib/asyncio/taskgroups.py b/Lib/asyncio/taskgroups.py
index 3ca65062efd272..ced9334df02a43 100644
--- a/Lib/asyncio/taskgroups.py
+++ b/Lib/asyncio/taskgroups.py
@@ -63,7 +63,7 @@ async def __aexit__(self, et, exc, tb):
 
         if et is not None:
             if et is exceptions.CancelledError:
-                if self._parent_cancel_requested and not self._parent_task.uncancel():
+                if self._parent_cancel_requested and tasks._uncancel(self._parent_task) == 0:
                     # Do nothing, i.e. swallow the error.
                     pass
                 else:
diff --git a/Lib/asyncio/tasks.py b/Lib/asyncio/tasks.py
index 3952b5f2a7743d..73881675195533 100644
--- a/Lib/asyncio/tasks.py
+++ b/Lib/asyncio/tasks.py
@@ -67,14 +67,30 @@ def _set_task_name(task, name):
         try:
             set_name = task.set_name
         except AttributeError:
-            warnings.warn("Task.set_name() was added in Python 3.8, "
+            warnings._deprecated(
+                name="missing asyncio.Task.set_name()",
+                message="Task.set_name() was added in Python 3.8, "
                       "the method support will be mandatory for third-party "
-                      "task implementations since 3.13.",
-                      DeprecationWarning, stacklevel=3)
+                      "task implementations since 3.13.", remove=(3, 13))
         else:
             set_name(name)
 
 
+def _uncancel(task):
+    try:
+        uncancel = task.uncancel
+    except AttributeError:
+        warnings._deprecated(
+            name="missing asyncio.Task.uncancel()",
+            message="Task.uncancel() was added in Python 3.11, "
+                  "the method support will be mandatory for third-party "
+                  "task implementations since 3.16.",
+                  remove=(3, 16))
+        return -1
+    else:
+        return uncancel()
+
+
 class Task(futures._PyFuture):  # Inherit Python Task implementation
                                 # from a Python Future implementation.
 
diff --git a/Lib/asyncio/timeouts.py b/Lib/asyncio/timeouts.py
index a89205348ff24c..85478ba93879c7 100644
--- a/Lib/asyncio/timeouts.py
+++ b/Lib/asyncio/timeouts.py
@@ -92,7 +92,7 @@ async def __aexit__(
         if self._state is _State.EXPIRING:
             self._state = _State.EXPIRED
 
-            if self._task.uncancel() == 0 and exc_type is exceptions.CancelledError:
+            if tasks._uncancel(self._task) == 0 and exc_type is exceptions.CancelledError:
                 # Since there are no outstanding cancel requests, we're
                 # handling this.
                 raise TimeoutError
diff --git a/Misc/NEWS.d/next/Library/2022-07-21-17-19-41.gh-issue-95098.eLTdR_.rst b/Misc/NEWS.d/next/Library/2022-07-21-17-19-41.gh-issue-95098.eLTdR_.rst
new file mode 100644
index 00000000000000..e2655af600679f
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2022-07-21-17-19-41.gh-issue-95098.eLTdR_.rst
@@ -0,0 +1 @@
+support custom tasks with missing uncancel methods