diff --git a/doc/whats_new/v1.3.rst b/doc/whats_new/v1.3.rst
index 2587e605cc5a5..bfddc45b91afd 100644
--- a/doc/whats_new/v1.3.rst
+++ b/doc/whats_new/v1.3.rst
@@ -480,6 +480,10 @@ Changelog
   when `n_neighbors` is large and `algorithm="brute"` with non Euclidean metrics.
   :pr:`24076` by :user:`Meekail Zain <micky774>`, :user:`Julien Jerphanion <jjerphan>`.
 
+- |API| The support for metrics other than `euclidean` and `manhattan` and for
+  callables in :class:`neighbors.NearestNeighbors` is deprecated and will be removed in
+  version 1.5. :pr:`24083` by :user:`Valentin Laurent <Valentin-Laurent>`.
+
 :mod:`sklearn.neural_network`
 .............................
 
diff --git a/sklearn/neighbors/_nearest_centroid.py b/sklearn/neighbors/_nearest_centroid.py
index 4e5ce354cc257..7b9c2479747d3 100644
--- a/sklearn/neighbors/_nearest_centroid.py
+++ b/sklearn/neighbors/_nearest_centroid.py
@@ -47,6 +47,11 @@ class NearestCentroid(ClassifierMixin, BaseEstimator):
         If the `"manhattan"` metric is provided, this centroid is the median
         and for all other metrics, the centroid is now set to be the mean.
 
+        .. deprecated:: 1.3
+            Support for metrics other than `euclidean` and `manhattan` and for
+            callables was deprecated in version 1.3 and will be removed in
+            version 1.5.
+
         .. versionchanged:: 0.19
             `metric='precomputed'` was deprecated and now raises an error
 
@@ -101,10 +106,12 @@ class NearestCentroid(ClassifierMixin, BaseEstimator):
     [1]
     """
 
+    _valid_metrics = set(_VALID_METRICS) - {"mahalanobis", "seuclidean", "wminkowski"}
+
     _parameter_constraints: dict = {
         "metric": [
             StrOptions(
-                set(_VALID_METRICS) - {"mahalanobis", "seuclidean", "wminkowski"}
+                _valid_metrics, deprecated=_valid_metrics - {"manhattan", "euclidean"}
             ),
             callable,
         ],
@@ -134,6 +141,20 @@ def fit(self, X, y):
             Fitted estimator.
         """
         self._validate_params()
+
+        if isinstance(self.metric, str) and self.metric not in (
+            "manhattan",
+            "euclidean",
+        ):
+            warnings.warn(
+                (
+                    "Support for distance metrics other than euclidean and "
+                    "manhattan and for callables was deprecated in version "
+                    "1.3 and will be removed in version 1.5."
+                ),
+                FutureWarning,
+            )
+
         # If X is sparse and the metric is "manhattan", store it in a csc
         # format is easier to calculate the median.
         if self.metric == "manhattan":
@@ -167,7 +188,6 @@ def fit(self, X, y):
             if is_X_sparse:
                 center_mask = np.where(center_mask)[0]
 
-            # XXX: Update other averaging methods according to the metrics.
             if self.metric == "manhattan":
                 # NumPy does not calculate median of sparse matrices.
                 if not is_X_sparse:
@@ -175,6 +195,7 @@ def fit(self, X, y):
                 else:
                     self.centroids_[cur_class] = csc_median_axis_0(X[center_mask])
             else:
+                # TODO(1.5) remove warning when metric is only manhattan or euclidean
                 if self.metric != "euclidean":
                     warnings.warn(
                         "Averaging for metrics other than "
@@ -209,6 +230,7 @@ def fit(self, X, y):
             self.centroids_ = dataset_centroid_[np.newaxis, :] + msd
         return self
 
+    # TODO(1.5) remove note about precomputed metric
     def predict(self, X):
         """Perform classification on an array of test vectors `X`.
 
diff --git a/sklearn/neighbors/tests/test_nearest_centroid.py b/sklearn/neighbors/tests/test_nearest_centroid.py
index 30ae1191c7ecf..861d09e92683c 100644
--- a/sklearn/neighbors/tests/test_nearest_centroid.py
+++ b/sklearn/neighbors/tests/test_nearest_centroid.py
@@ -53,6 +53,8 @@ def test_classification_toy():
     assert_array_equal(clf.predict(T_csr.tolil()), true_result)
 
 
+# TODO(1.5): Remove filterwarnings when support for some metrics is removed
+@pytest.mark.filterwarnings("ignore:Support for distance metrics:FutureWarning:sklearn")
 def test_iris():
     # Check consistency on dataset iris.
     for metric in ("euclidean", "cosine"):
@@ -61,6 +63,8 @@ def test_iris():
         assert score > 0.9, "Failed with score = " + str(score)
 
 
+# TODO(1.5): Remove filterwarnings when support for some metrics is removed
+@pytest.mark.filterwarnings("ignore:Support for distance metrics:FutureWarning:sklearn")
 def test_iris_shrinkage():
     # Check consistency on dataset iris, when using shrinkage.
     for metric in ("euclidean", "cosine"):
@@ -142,6 +146,20 @@ def test_manhattan_metric():
     assert_array_equal(dense_centroid, [[-1, -1], [1, 1]])
 
 
+# TODO(1.5): remove this test
+@pytest.mark.parametrize(
+    "metric", sorted(list(NearestCentroid._valid_metrics - {"manhattan", "euclidean"}))
+)
+def test_deprecated_distance_metric_supports(metric):
+    # Check that a warning is raised for all deprecated distance metric supports
+    clf = NearestCentroid(metric=metric)
+    with pytest.warns(
+        FutureWarning,
+        match="Support for distance metrics other than euclidean and manhattan",
+    ):
+        clf.fit(X, y)
+
+
 def test_features_zero_var():
     # Test that features with 0 variance throw error