diff --git a/sklearn/metrics/_pairwise_distances_reduction/_datasets_pair.pyx.tp b/sklearn/metrics/_pairwise_distances_reduction/_datasets_pair.pyx.tp
index 8899f49330440..2c3ca44047145 100644
--- a/sklearn/metrics/_pairwise_distances_reduction/_datasets_pair.pyx.tp
+++ b/sklearn/metrics/_pairwise_distances_reduction/_datasets_pair.pyx.tp
@@ -1,3 +1,5 @@
+import copy
+
 {{py:
 
 implementation_specific_values = [
@@ -84,12 +86,17 @@ cdef class DatasetsPair{{name_suffix}}:
         datasets_pair: DatasetsPair{{name_suffix}}
             The suited DatasetsPair{{name_suffix}} implementation.
         """
-        # Y_norm_squared might be propagated down to DatasetsPairs
-        # via metrics_kwargs when the Euclidean specialisations
-        # can't be used. To prevent Y_norm_squared to be passed
+        # X_norm_squared and Y_norm_squared might be propagated
+        # down to DatasetsPairs via metrics_kwargs when the Euclidean
+        # specialisations can't be used.
+        # To prevent X_norm_squared and Y_norm_squared to be passed
         # down to DistanceMetrics (whose constructors would raise
-        # a RuntimeError), we pop it here.
+        # a RuntimeError), we pop them here.
         if metric_kwargs is not None:
+            # Copying metric_kwargs not to pop "X_norm_squared"
+            # and "Y_norm_squared" where they are used
+            metric_kwargs = copy.copy(metric_kwargs)
+            metric_kwargs.pop("X_norm_squared", None)
             metric_kwargs.pop("Y_norm_squared", None)
         cdef:
             {{DistanceMetric}} distance_metric = DistanceMetric.get_metric(