diff --git a/sklearn/cluster/_spectral.py b/sklearn/cluster/_spectral.py
index d3cc2979d4af2..dc70d6a515476 100644
--- a/sklearn/cluster/_spectral.py
+++ b/sklearn/cluster/_spectral.py
@@ -744,7 +744,7 @@ def fit(self, X, y=None):
                 params["coef0"] = self.coef0
             self.affinity_matrix_ = pairwise_kernels(
                 X, metric=self.affinity, filter_params=True, **params
-            )
+            ).astype(X.dtype, copy=False)
 
         random_state = check_random_state(self.random_state)
         self.labels_ = spectral_clustering(
@@ -787,5 +787,6 @@ def fit_predict(self, X, y=None):
     def _more_tags(self):
         return {
             "pairwise": self.affinity
-            in ["precomputed", "precomputed_nearest_neighbors"]
+            in ["precomputed", "precomputed_nearest_neighbors"],
+            "preserves_dtype": [np.float64, np.float32],
         }
diff --git a/sklearn/cluster/tests/test_spectral.py b/sklearn/cluster/tests/test_spectral.py
index ea40bc7c79139..5fd181e9efae7 100644
--- a/sklearn/cluster/tests/test_spectral.py
+++ b/sklearn/cluster/tests/test_spectral.py
@@ -40,7 +40,7 @@
 
 @pytest.mark.parametrize("eigen_solver", ("arpack", "lobpcg"))
 @pytest.mark.parametrize("assign_labels", ("kmeans", "discretize", "cluster_qr"))
-def test_spectral_clustering(eigen_solver, assign_labels):
+def test_spectral_clustering(eigen_solver, assign_labels, global_dtype):
     S = np.array(
         [
             [1.0, 1.0, 1.0, 0.2, 0.0, 0.0, 0.0],
@@ -50,7 +50,8 @@ def test_spectral_clustering(eigen_solver, assign_labels):
             [0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 1.0],
             [0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 1.0],
             [0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 1.0],
-        ]
+        ],
+        dtype=global_dtype,
     )
 
     for mat in (S, sparse.csr_matrix(S)):
@@ -74,14 +75,14 @@ def test_spectral_clustering(eigen_solver, assign_labels):
 
 
 @pytest.mark.parametrize("assign_labels", ("kmeans", "discretize", "cluster_qr"))
-def test_spectral_clustering_sparse(assign_labels):
+def test_spectral_clustering_sparse(assign_labels, global_dtype):
     X, y = make_blobs(
         n_samples=20, random_state=0, centers=[[1, 1], [-1, -1]], cluster_std=0.01
     )
 
     S = rbf_kernel(X, gamma=1)
     S = np.maximum(S - 1e-4, 0)
-    S = sparse.coo_matrix(S)
+    S = sparse.coo_matrix(S, dtype=global_dtype)
 
     labels = (
         SpectralClustering(
@@ -96,11 +97,12 @@ def test_spectral_clustering_sparse(assign_labels):
     assert adjusted_rand_score(y, labels) == 1
 
 
-def test_precomputed_nearest_neighbors_filtering():
+def test_precomputed_nearest_neighbors_filtering(global_dtype):
     # Test precomputed graph filtering when containing too many neighbors
     X, y = make_blobs(
         n_samples=200, random_state=0, centers=[[1, 1], [-1, -1]], cluster_std=0.01
     )
+    X = X.astype(global_dtype, copy=False)
 
     n_neighbors = 2
     results = []
@@ -122,13 +124,14 @@ def test_precomputed_nearest_neighbors_filtering():
     assert_array_equal(results[0], results[1])
 
 
-def test_affinities():
+def test_affinities(global_dtype):
     # Note: in the following, random_state has been selected to have
     # a dataset that yields a stable eigen decomposition both when built
     # on OSX and Linux
     X, y = make_blobs(
         n_samples=20, random_state=0, centers=[[1, 1], [-1, -1]], cluster_std=0.01
     )
+    X = X.astype(global_dtype, copy=False)
     # nearest neighbors affinity
     sp = SpectralClustering(n_clusters=2, affinity="nearest_neighbors", random_state=0)
     with pytest.warns(UserWarning, match="not fully connected"):
@@ -140,6 +143,7 @@ def test_affinities():
     assert adjusted_rand_score(y, labels) == 1
 
     X = check_random_state(10).rand(10, 5) * 10
+    X = X.astype(global_dtype, copy=False)
 
     kernels_available = kernel_metrics()
     for kern in kernels_available:
@@ -182,13 +186,13 @@ def test_cluster_qr():
     assert np.array_equal(labels_float64, labels_float32)
 
 
-def test_cluster_qr_permutation_invariance():
+def test_cluster_qr_permutation_invariance(global_dtype):
     # cluster_qr must be invariant to sample permutation.
     random_state = np.random.RandomState(seed=8)
     n_samples, n_components = 100, 5
-    data = random_state.randn(n_samples, n_components)
+    data = random_state.randn(n_samples, n_components).astype(global_dtype, copy=False)
     perm = random_state.permutation(n_samples)
-    assert np.array_equal(
+    assert_array_equal(
         cluster_qr(data)[perm],
         cluster_qr(data[perm]),
     )
@@ -263,12 +267,13 @@ def test_spectral_clustering_with_arpack_amg_solvers():
             spectral_clustering(graph, n_clusters=2, eigen_solver="amg", random_state=0)
 
 
-def test_n_components():
+def test_n_components(global_dtype):
     # Test that after adding n_components, result is different and
     # n_components = n_clusters by default
     X, y = make_blobs(
         n_samples=20, random_state=0, centers=[[1, 1], [-1, -1]], cluster_std=0.01
     )
+    X = X.astype(global_dtype, copy=False)
     sp = SpectralClustering(n_clusters=2, random_state=0)
     labels = sp.fit(X).labels_
     # set n_components = n_cluster and test if result is the same