diff --git a/distributed/client.py b/distributed/client.py
index b21a0504352..ef14a39b0cb 100644
--- a/distributed/client.py
+++ b/distributed/client.py
@@ -60,6 +60,7 @@
     CommClosedError,
     ConnectionPool,
     PooledRPCCall,
+    Status,
     clean_exception,
     connect,
     rpc,
@@ -825,6 +826,11 @@ def __init__(
         elif isinstance(getattr(address, "scheduler_address", None), str):
             # It's a LocalCluster or LocalCluster-compatible object
             self.cluster = address
+            status = getattr(self.cluster, "status")
+            if status and status in [Status.closed, Status.closing]:
+                raise RuntimeError(
+                    f"Trying to connect to an already closed or closing Cluster {self.cluster}."
+                )
             with suppress(AttributeError):
                 loop = address.loop
             if security is None:
diff --git a/distributed/deploy/tests/test_local.py b/distributed/deploy/tests/test_local.py
index ae1a96779b1..1d6cd954be7 100644
--- a/distributed/deploy/tests/test_local.py
+++ b/distributed/deploy/tests/test_local.py
@@ -1149,3 +1149,18 @@ async def test_cluster_host_used_throughout_cluster(host, use_nanny):
             if use_nanny:
                 url = urlparse(worker.process.worker_address)
                 assert url.hostname == "127.0.0.1"
+
+
+@gen_test()
+async def test_connect_to_closed_cluster(cleanup):
+    async with LocalCluster(processes=False, asynchronous=True) as cluster:
+        async with Client(cluster, asynchronous=True) as c1:
+            assert await c1.submit(inc, 1) == 2
+
+    with pytest.raises(
+        RuntimeError,
+        match="Trying to connect to an already closed or closing Cluster",
+    ):
+        # Raises during init without actually connecting since we're not
+        # awaiting anything
+        Client(cluster, asynchronous=True)