diff --git a/doc/whats_new/v1.1.rst b/doc/whats_new/v1.1.rst
index 61e074e56a657..ef2308997f898 100644
--- a/doc/whats_new/v1.1.rst
+++ b/doc/whats_new/v1.1.rst
@@ -95,6 +95,10 @@ Changelog
   See :func:`cluster.spectral_clustering` for more details.
   :pr:`21148` by :user:`Andrew Knyazev <lobpcg>`
 
+- |Enhancement| Adds :term:`get_feature_names_out` to :class:`cluster.Birch`,
+  :class:`cluster.FeatureAgglomeration`, :class:`cluster.KMeans`,
+  :class:`cluster.MiniBatchKMeans`. :pr:`22255` by `Thomas Fan`_.
+
 - |Efficiency| In :class:`cluster.KMeans`, the default ``algorithm`` is now
   ``"lloyd"`` which is the full classical EM-style algorithm. Both ``"auto"``
   and ``"full"`` are deprecated and will be removed in version 1.3. They are
diff --git a/sklearn/cluster/_agglomerative.py b/sklearn/cluster/_agglomerative.py
index 4bc49ea2301e6..68ab834202753 100644
--- a/sklearn/cluster/_agglomerative.py
+++ b/sklearn/cluster/_agglomerative.py
@@ -14,7 +14,7 @@
 from scipy import sparse
 from scipy.sparse.csgraph import connected_components
 
-from ..base import BaseEstimator, ClusterMixin
+from ..base import BaseEstimator, ClusterMixin, _ClassNamePrefixFeaturesOutMixin
 from ..metrics.pairwise import paired_distances
 from ..metrics import DistanceMetric
 from ..metrics._dist_metrics import METRIC_MAPPING
@@ -1054,7 +1054,9 @@ def fit_predict(self, X, y=None):
         return super().fit_predict(X, y)
 
 
-class FeatureAgglomeration(AgglomerativeClustering, AgglomerationTransform):
+class FeatureAgglomeration(
+    _ClassNamePrefixFeaturesOutMixin, AgglomerativeClustering, AgglomerationTransform
+):
     """Agglomerate features.
 
     Recursively merges pair of clusters of features.
@@ -1236,6 +1238,7 @@ def fit(self, X, y=None):
         """
         X = self._validate_data(X, ensure_min_features=2)
         super()._fit(X.T)
+        self._n_features_out = self.n_clusters_
         return self
 
     @property
diff --git a/sklearn/cluster/_birch.py b/sklearn/cluster/_birch.py
index 8e86d8dd6ba08..3e47cc7b74492 100644
--- a/sklearn/cluster/_birch.py
+++ b/sklearn/cluster/_birch.py
@@ -11,7 +11,12 @@
 
 from ..metrics import pairwise_distances_argmin
 from ..metrics.pairwise import euclidean_distances
-from ..base import TransformerMixin, ClusterMixin, BaseEstimator
+from ..base import (
+    TransformerMixin,
+    ClusterMixin,
+    BaseEstimator,
+    _ClassNamePrefixFeaturesOutMixin,
+)
 from ..utils.extmath import row_norms
 from ..utils import check_scalar, deprecated
 from ..utils.validation import check_is_fitted
@@ -342,7 +347,9 @@ def radius(self):
         return sqrt(max(0, sq_radius))
 
 
-class Birch(ClusterMixin, TransformerMixin, BaseEstimator):
+class Birch(
+    _ClassNamePrefixFeaturesOutMixin, ClusterMixin, TransformerMixin, BaseEstimator
+):
     """Implements the BIRCH clustering algorithm.
 
     It is a memory-efficient, online-learning algorithm provided as an
@@ -599,6 +606,7 @@ def _fit(self, X, partial):
 
         centroids = np.concatenate([leaf.centroids_ for leaf in self._get_leaves()])
         self.subcluster_centers_ = centroids
+        self._n_features_out = self.subcluster_centers_.shape[0]
 
         self._global_clustering(X)
         return self
diff --git a/sklearn/cluster/_kmeans.py b/sklearn/cluster/_kmeans.py
index b631b1f77b26a..51d87044f5496 100644
--- a/sklearn/cluster/_kmeans.py
+++ b/sklearn/cluster/_kmeans.py
@@ -16,7 +16,12 @@
 import numpy as np
 import scipy.sparse as sp
 
-from ..base import BaseEstimator, ClusterMixin, TransformerMixin
+from ..base import (
+    BaseEstimator,
+    ClusterMixin,
+    TransformerMixin,
+    _ClassNamePrefixFeaturesOutMixin,
+)
 from ..metrics.pairwise import euclidean_distances
 from ..metrics.pairwise import _euclidean_distances
 from ..utils.extmath import row_norms, stable_cumsum
@@ -767,7 +772,9 @@ def _labels_inertia_threadpool_limit(
     return labels, inertia
 
 
-class KMeans(TransformerMixin, ClusterMixin, BaseEstimator):
+class KMeans(
+    _ClassNamePrefixFeaturesOutMixin, TransformerMixin, ClusterMixin, BaseEstimator
+):
     """K-Means clustering.
 
     Read more in the :ref:`User Guide <k_means>`.
@@ -1240,6 +1247,7 @@ def fit(self, X, y=None, sample_weight=None):
             )
 
         self.cluster_centers_ = best_centers
+        self._n_features_out = self.cluster_centers_.shape[0]
         self.labels_ = best_labels
         self.inertia_ = best_inertia
         self.n_iter_ = best_n_iter
@@ -2020,6 +2028,7 @@ def fit(self, X, y=None, sample_weight=None):
                     break
 
         self.cluster_centers_ = centers
+        self._n_features_out = self.cluster_centers_.shape[0]
 
         self.n_steps_ = i + 1
         self.n_iter_ = int(np.ceil(((i + 1) * self._batch_size) / n_samples))
@@ -2134,6 +2143,7 @@ def partial_fit(self, X, y=None, sample_weight=None):
             )
 
         self.n_steps_ += 1
+        self._n_features_out = self.cluster_centers_.shape[0]
 
         return self
 
diff --git a/sklearn/cluster/tests/test_birch.py b/sklearn/cluster/tests/test_birch.py
index 5d8a3222ef156..4e64524e2cb11 100644
--- a/sklearn/cluster/tests/test_birch.py
+++ b/sklearn/cluster/tests/test_birch.py
@@ -219,3 +219,14 @@ def test_birch_params_validation(params, err_type, err_msg):
     X, _ = make_blobs(n_samples=80, centers=4)
     with pytest.raises(err_type, match=err_msg):
         Birch(**params).fit(X)
+
+
+def test_feature_names_out():
+    """Check `get_feature_names_out` for `Birch`."""
+    X, _ = make_blobs(n_samples=80, n_features=4, random_state=0)
+    brc = Birch(n_clusters=4)
+    brc.fit(X)
+    n_clusters = brc.subcluster_centers_.shape[0]
+
+    names_out = brc.get_feature_names_out()
+    assert_array_equal([f"birch{i}" for i in range(n_clusters)], names_out)
diff --git a/sklearn/cluster/tests/test_feature_agglomeration.py b/sklearn/cluster/tests/test_feature_agglomeration.py
index 6d9a942e3dcfe..1f61093a9568d 100644
--- a/sklearn/cluster/tests/test_feature_agglomeration.py
+++ b/sklearn/cluster/tests/test_feature_agglomeration.py
@@ -4,8 +4,11 @@
 # Authors: Sergul Aydore 2017
 import numpy as np
 import pytest
+
+from numpy.testing import assert_array_equal
 from sklearn.cluster import FeatureAgglomeration
 from sklearn.utils._testing import assert_array_almost_equal
+from sklearn.datasets import make_blobs
 
 
 def test_feature_agglomeration():
@@ -41,3 +44,16 @@ def test_feature_agglomeration():
 
     assert_array_almost_equal(agglo_mean.transform(X_full_mean), Xt_mean)
     assert_array_almost_equal(agglo_median.transform(X_full_median), Xt_median)
+
+
+def test_feature_agglomeration_feature_names_out():
+    """Check `get_feature_names_out` for `FeatureAgglomeration`."""
+    X, _ = make_blobs(n_features=6, random_state=0)
+    agglo = FeatureAgglomeration(n_clusters=3)
+    agglo.fit(X)
+    n_clusters = agglo.n_clusters_
+
+    names_out = agglo.get_feature_names_out()
+    assert_array_equal(
+        [f"featureagglomeration{i}" for i in range(n_clusters)], names_out
+    )
diff --git a/sklearn/cluster/tests/test_k_means.py b/sklearn/cluster/tests/test_k_means.py
index 6e395778418c8..2d62aaaba96e9 100644
--- a/sklearn/cluster/tests/test_k_means.py
+++ b/sklearn/cluster/tests/test_k_means.py
@@ -1205,3 +1205,18 @@ def test_is_same_clustering():
     # mapped to a same value
     labels3 = np.array([1, 0, 0, 2, 2, 0, 2, 1], dtype=np.int32)
     assert not _is_same_clustering(labels1, labels3, 3)
+
+
+@pytest.mark.parametrize(
+    "Klass, method",
+    [(KMeans, "fit"), (MiniBatchKMeans, "fit"), (MiniBatchKMeans, "partial_fit")],
+)
+def test_feature_names_out(Klass, method):
+    """Check `feature_names_out` for `KMeans` and `MiniBatchKMeans`."""
+    class_name = Klass.__name__.lower()
+    kmeans = Klass()
+    getattr(kmeans, method)(X)
+    n_clusters = kmeans.cluster_centers_.shape[0]
+
+    names_out = kmeans.get_feature_names_out()
+    assert_array_equal([f"{class_name}{i}" for i in range(n_clusters)], names_out)
diff --git a/sklearn/tests/test_common.py b/sklearn/tests/test_common.py
index a8178a4219485..7f6f99dcc8b67 100644
--- a/sklearn/tests/test_common.py
+++ b/sklearn/tests/test_common.py
@@ -380,7 +380,6 @@ def test_pandas_column_name_consistency(estimator):
 # TODO: As more modules support get_feature_names_out they should be removed
 # from this list to be tested
 GET_FEATURES_OUT_MODULES_TO_IGNORE = [
-    "cluster",
     "ensemble",
     "isotonic",
     "kernel_approximation",