diff --git a/distributed/client.py b/distributed/client.py
index b2e227b9604..c4abed3e29b 100644
--- a/distributed/client.py
+++ b/distributed/client.py
@@ -755,7 +755,11 @@ async def done_callback(future, callback):
     """
     while future.status == "pending":
         await future._state.wait()
-    callback(future)
+    try:
+        callback(future)
+    except RuntimeError as e:
+        if "shutdown" not in str(e) and "interpreter" not in str(e):
+            raise
 
 
 class AllExit(Exception):
diff --git a/distributed/tests/test_client.py b/distributed/tests/test_client.py
index fbb08076613..3a1f08cfead 100644
--- a/distributed/tests/test_client.py
+++ b/distributed/tests/test_client.py
@@ -8435,3 +8435,24 @@ def reducer(futs, *, offset=0, **kwargs):
 
     result = await future.result()
     assert result == 30 if not offset else 31
+
+
+@gen_cluster(client=True)
+async def test_done_callback_shutdown_runtime_error(c, s, a, b):
+    from distributed.client import done_callback
+
+    future = c.submit(inc, 1)
+    await future
+
+    # Shutdown-related RuntimeError is swallowed
+    def cb_shutdown(fut):
+        raise RuntimeError("cannot schedule new futures after interpreter shutdown")
+
+    await done_callback(future, cb_shutdown)
+
+    # Unrelated RuntimeError is re-raised
+    def cb_other(fut):
+        raise RuntimeError("something else")
+
+    with pytest.raises(RuntimeError, match="something else"):
+        await done_callback(future, cb_other)