diff --git a/src/lightning/fabric/utilities/cloud_io.py b/src/lightning/fabric/utilities/cloud_io.py
index 9d0a33afd0b77..637dfcd9b1671 100644
--- a/src/lightning/fabric/utilities/cloud_io.py
+++ b/src/lightning/fabric/utilities/cloud_io.py
@@ -13,6 +13,7 @@
 # limitations under the License.
 """Utilities related to data saving/loading."""
 
+import errno
 import io
 import logging
 from pathlib import Path
@@ -84,10 +85,16 @@ def _atomic_save(checkpoint: dict[str, Any], filepath: Union[str, Path]) -> None
     log.debug(f"Saving checkpoint: {filepath}")
     torch.save(checkpoint, bytesbuffer)
 
-    # We use a transaction here to avoid file corruption if the save gets interrupted
-    fs, urlpath = fsspec.core.url_to_fs(str(filepath))
-    with fs.transaction, fs.open(urlpath, "wb") as f:
-        f.write(bytesbuffer.getvalue())
+    try:
+        # We use a transaction here to avoid file corruption if the save gets interrupted
+        fs, urlpath = fsspec.core.url_to_fs(str(filepath))
+        with fs.transaction, fs.open(urlpath, "wb") as f:
+            f.write(bytesbuffer.getvalue())
+    except PermissionError as e:
+        if isinstance(e.__context__, OSError) and getattr(e.__context__, "errno", None) == errno.EXDEV:
+            raise RuntimeError(
+                'Upgrade fsspec to enable cross-device local checkpoints: pip install "fsspec[http]>=2025.5.0"',
+            ) from e
 
 
 def _is_object_storage(fs: AbstractFileSystem) -> bool:
diff --git a/src/lightning/pytorch/CHANGELOG.md b/src/lightning/pytorch/CHANGELOG.md
index 0095367e9187a..18ef679312a66 100644
--- a/src/lightning/pytorch/CHANGELOG.md
+++ b/src/lightning/pytorch/CHANGELOG.md
@@ -11,12 +11,13 @@ The format is based on [Keep a Changelog](http://keepachangelog.com/en/1.0.0/).
 
 - Add enable_autolog_hparams argument to Trainer ([#20593](https://github.com/Lightning-AI/pytorch-lightning/pull/20593))
 
+- For cross-device local checkpoints, instruct users to install `fsspec>=2025.5.0` if unavailable ([#20780](https://github.com/Lightning-AI/pytorch-lightning/pull/20780))
+
 
 ### Changed
 
 -
 
-
 ### Removed
 
 -
diff --git a/src/lightning/pytorch/callbacks/model_checkpoint.py b/src/lightning/pytorch/callbacks/model_checkpoint.py
index 85bfb65c0ea6e..6b7b2831a2e04 100644
--- a/src/lightning/pytorch/callbacks/model_checkpoint.py
+++ b/src/lightning/pytorch/callbacks/model_checkpoint.py
@@ -155,6 +155,10 @@ class ModelCheckpoint(Checkpoint):
         If the checkpoint's ``dirpath`` changed from what it was before while resuming the training,
         only ``best_model_path`` will be reloaded and a warning will be issued.
 
+        If you provide a ``filename`` on a mounted device where changing permissions is not allowed (causing ``chmod``
+        to raise a ``PermissionError``), install `fsspec>=2025.5.0`. Then the error is caught, the file's permissions
+        remain unchanged, and the checkpoint is still saved. Otherwise, no checkpoint will be saved and training stops.
+
     Raises:
         MisconfigurationException:
             If ``save_top_k`` is smaller than ``-1``,
diff --git a/tests/tests_pytorch/trainer/connectors/test_checkpoint_connector.py b/tests/tests_pytorch/trainer/connectors/test_checkpoint_connector.py
index 722742a3ccae0..662fd99d1b12c 100644
--- a/tests/tests_pytorch/trainer/connectors/test_checkpoint_connector.py
+++ b/tests/tests_pytorch/trainer/connectors/test_checkpoint_connector.py
@@ -11,10 +11,13 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
+import errno
 import os
+import re
 from unittest import mock
 from unittest.mock import ANY, Mock
 
+import fsspec
 import pytest
 import torch
 
@@ -105,6 +108,31 @@ def test_hpc_max_ckpt_version(tmp_path):
     )
 
 
+def test_local_cross_device_checkpoint(tmpdir):
+    """Test that the _CheckpointConnector can write local cross-device files or raises an error if fsspec<2025.5.0."""
+    model = BoringModel()
+    # hardcoding dir since `tmp_path` can be windows path
+    trainer = Trainer(
+        default_root_dir="memory://test_ckpt_for_fsspec", limit_train_batches=1, limit_val_batches=1, max_epochs=1
+    )
+    trainer.fit(model)
+    # Simulate the behavior of fsspec when writing to a local file system but other device.
+    with (
+        mock.patch("os.rename", side_effect=OSError(errno.EXDEV, "Invalid cross-device link")),
+        mock.patch("os.chmod", side_effect=PermissionError("Operation not permitted")),
+    ):
+        if fsspec.__version__ < "2025.5.0":
+            with pytest.raises(
+                RuntimeError,
+                match=re.escape(
+                    'Upgrade fsspec to enable cross-device local checkpoints: pip install "fsspec[http]>=2025.5.0"'
+                ),
+            ):
+                trainer.save_checkpoint(tmpdir + "/test_ckpt_for_fsspec/hpc_ckpt.ckpt")
+        else:
+            trainer.save_checkpoint(tmpdir + "/test_ckpt_for_fsspec/hpc_ckpt.ckpt")
+
+
 def test_ckpt_for_fsspec():
     """Test that the _CheckpointConnector is able to write to fsspec file systems."""
     model = BoringModel()