diff --git a/sklearn/metrics/_pairwise_distances_reduction/_argkmin.pyx.tp b/sklearn/metrics/_pairwise_distances_reduction/_argkmin.pyx.tp
index eec2e2aabdd06..b8afe5c3cd5f8 100644
--- a/sklearn/metrics/_pairwise_distances_reduction/_argkmin.pyx.tp
+++ b/sklearn/metrics/_pairwise_distances_reduction/_argkmin.pyx.tp
@@ -330,11 +330,8 @@ cdef class EuclideanArgKmin{{name_suffix}}(ArgKmin{{name_suffix}}):
         metric_kwargs=None,
     ):
         if (
-            metric_kwargs is not None and
-            len(metric_kwargs) > 0 and (
-                "Y_norm_squared" not in metric_kwargs or
-                "X_norm_squared" not in metric_kwargs
-            )
+            isinstance(metric_kwargs, dict) and
+            (metric_kwargs.keys() - {"X_norm_squared", "Y_norm_squared"})
         ):
             warnings.warn(
                 f"Some metric_kwargs have been passed ({metric_kwargs}) but aren't "
diff --git a/sklearn/metrics/_pairwise_distances_reduction/_radius_neighbors.pyx.tp b/sklearn/metrics/_pairwise_distances_reduction/_radius_neighbors.pyx.tp
index 0fdc3bb50203f..b3f20cac3ea08 100644
--- a/sklearn/metrics/_pairwise_distances_reduction/_radius_neighbors.pyx.tp
+++ b/sklearn/metrics/_pairwise_distances_reduction/_radius_neighbors.pyx.tp
@@ -336,11 +336,8 @@ cdef class EuclideanRadiusNeighbors{{name_suffix}}(RadiusNeighbors{{name_suffix}
         metric_kwargs=None,
     ):
         if (
-            metric_kwargs is not None and
-            len(metric_kwargs) > 0 and (
-                "Y_norm_squared" not in metric_kwargs or
-                "X_norm_squared" not in metric_kwargs
-            )
+            isinstance(metric_kwargs, dict) and
+            (metric_kwargs.keys() - {"X_norm_squared", "Y_norm_squared"})
         ):
             warnings.warn(
                 f"Some metric_kwargs have been passed ({metric_kwargs}) but aren't "
diff --git a/sklearn/metrics/tests/test_pairwise_distances_reduction.py b/sklearn/metrics/tests/test_pairwise_distances_reduction.py
index c334087c65448..4fe8013cd3602 100644
--- a/sklearn/metrics/tests/test_pairwise_distances_reduction.py
+++ b/sklearn/metrics/tests/test_pairwise_distances_reduction.py
@@ -1,5 +1,6 @@
 import itertools
 import re
+import warnings
 from collections import defaultdict
 
 import numpy as np
@@ -620,19 +621,44 @@ def test_argkmin_factory_method_wrong_usages():
     with pytest.raises(ValueError, match="ndarray is not C-contiguous"):
         ArgKmin.compute(X=np.asfortranarray(X), Y=Y, k=k, metric=metric)
 
+    # A UserWarning must be raised in this case.
     unused_metric_kwargs = {"p": 3}
 
-    message = (
-        r"Some metric_kwargs have been passed \({'p': 3}\) but aren't usable for this"
-        r" case \("
-        r"EuclideanArgKmin64."
-    )
+    message = r"Some metric_kwargs have been passed \({'p': 3}\) but"
 
     with pytest.warns(UserWarning, match=message):
         ArgKmin.compute(
             X=X, Y=Y, k=k, metric=metric, metric_kwargs=unused_metric_kwargs
         )
 
+    # A UserWarning must be raised in this case.
+    metric_kwargs = {
+        "p": 3,  # unused
+        "Y_norm_squared": sqeuclidean_row_norms(Y, num_threads=2),
+    }
+
+    message = r"Some metric_kwargs have been passed \({'p': 3, 'Y_norm_squared'"
+
+    with pytest.warns(UserWarning, match=message):
+        ArgKmin.compute(X=X, Y=Y, k=k, metric=metric, metric_kwargs=metric_kwargs)
+
+    # No user warning must be raised in this case.
+    metric_kwargs = {
+        "X_norm_squared": sqeuclidean_row_norms(X, num_threads=2),
+    }
+    with warnings.catch_warnings():
+        warnings.simplefilter("error", category=UserWarning)
+        ArgKmin.compute(X=X, Y=Y, k=k, metric=metric, metric_kwargs=metric_kwargs)
+
+    # No user warning must be raised in this case.
+    metric_kwargs = {
+        "X_norm_squared": sqeuclidean_row_norms(X, num_threads=2),
+        "Y_norm_squared": sqeuclidean_row_norms(Y, num_threads=2),
+    }
+    with warnings.catch_warnings():
+        warnings.simplefilter("error", category=UserWarning)
+        ArgKmin.compute(X=X, Y=Y, k=k, metric=metric, metric_kwargs=metric_kwargs)
+
 
 def test_radius_neighbors_factory_method_wrong_usages():
     rng = np.random.RandomState(1)
@@ -683,16 +709,48 @@ def test_radius_neighbors_factory_method_wrong_usages():
 
     unused_metric_kwargs = {"p": 3}
 
-    message = (
-        r"Some metric_kwargs have been passed \({'p': 3}\) but aren't usable for this"
-        r" case \(EuclideanRadiusNeighbors64"
-    )
+    # A UserWarning must be raised in this case.
+    message = r"Some metric_kwargs have been passed \({'p': 3}\) but"
 
     with pytest.warns(UserWarning, match=message):
         RadiusNeighbors.compute(
             X=X, Y=Y, radius=radius, metric=metric, metric_kwargs=unused_metric_kwargs
         )
 
+    # A UserWarning must be raised in this case.
+    metric_kwargs = {
+        "p": 3,  # unused
+        "Y_norm_squared": sqeuclidean_row_norms(Y, num_threads=2),
+    }
+
+    message = r"Some metric_kwargs have been passed \({'p': 3, 'Y_norm_squared'"
+
+    with pytest.warns(UserWarning, match=message):
+        RadiusNeighbors.compute(
+            X=X, Y=Y, radius=radius, metric=metric, metric_kwargs=metric_kwargs
+        )
+
+    # No user warning must be raised in this case.
+    metric_kwargs = {
+        "X_norm_squared": sqeuclidean_row_norms(X, num_threads=2),
+        "Y_norm_squared": sqeuclidean_row_norms(Y, num_threads=2),
+    }
+    with warnings.catch_warnings():
+        warnings.simplefilter("error", category=UserWarning)
+        RadiusNeighbors.compute(
+            X=X, Y=Y, radius=radius, metric=metric, metric_kwargs=metric_kwargs
+        )
+
+    # No user warning must be raised in this case.
+    metric_kwargs = {
+        "X_norm_squared": sqeuclidean_row_norms(X, num_threads=2),
+    }
+    with warnings.catch_warnings():
+        warnings.simplefilter("error", category=UserWarning)
+        RadiusNeighbors.compute(
+            X=X, Y=Y, radius=radius, metric=metric, metric_kwargs=metric_kwargs
+        )
+
 
 @pytest.mark.parametrize(
     "n_samples_X, n_samples_Y", [(100, 100), (500, 100), (100, 500)]