diff --git a/distributed/tests/test_scheduler.py b/distributed/tests/test_scheduler.py
index 1e1a3e69b32..f93499fb421 100644
--- a/distributed/tests/test_scheduler.py
+++ b/distributed/tests/test_scheduler.py
@@ -3218,3 +3218,29 @@ async def test_set_restrictions(c, s, a, b):
     assert s.tasks[f.key].worker_restrictions == {a.address}
     s.reschedule(f)
     await f
+
+
+@gen_cluster(
+    client=True,
+    nthreads=[("", 1, {}), ("", 1, {"memory_pause_fraction": 1e-15}), ("", 1, {})],
+)
+async def test_avoid_paused_workers(c, s, w1, w2, w3):
+    while s.workers[w2.address].status != Status.paused:
+        await asyncio.sleep(0.01)
+    futures = c.map(slowinc, range(8), delay=0.1)
+    while (len(w1.tasks), len(w2.tasks), len(w3.tasks)) != (4, 0, 4):
+        await asyncio.sleep(0.01)
+
+
+@gen_cluster(client=True, nthreads=[("", 1, {"memory_pause_fraction": 1e-15})])
+async def test_unpause_schedules_unrannable_tasks(c, s, a):
+    while s.workers[a.address].status != Status.paused:
+        await asyncio.sleep(0.01)
+
+    fut = c.submit(inc, 1, key="x")
+    while not s.unrunnable:
+        await asyncio.sleep(.001)
+    assert next(iter(s.unrunnable)).key == "x"
+
+    a.memory_pause_fraction = 0.8
+    assert await fut == 2
diff --git a/distributed/utils_test.py b/distributed/utils_test.py
index 3a72ed07889..e56e3865c77 100644
--- a/distributed/utils_test.py
+++ b/distributed/utils_test.py
@@ -69,7 +69,7 @@
     sync,
     thread_state,
 )
-from .worker import Worker
+from .worker import RUNNING, Worker
 
 try:
     import dask.array  # register config
@@ -1549,7 +1549,7 @@ def check_instances():
     for w in Worker._instances:
         with suppress(RuntimeError):  # closed IOLoop
             w.loop.add_callback(w.close, report=False, executor_wait=False)
-            if w.status in (Status.running, Status.paused):
+            if w.status in RUNNING:
                 w.loop.add_callback(w.close)
     Worker._instances.clear()
 
diff --git a/distributed/worker.py b/distributed/worker.py
index 3d9c9504054..06d914e3838 100644
--- a/distributed/worker.py
+++ b/distributed/worker.py
@@ -3673,11 +3673,7 @@ def get_worker() -> Worker:
         return thread_state.execution_state["worker"]
     except AttributeError:
         try:
-            return first(
-                w
-                for w in Worker._instances
-                if w.status in (Status.running, Status.paused)
-            )
+            return first(w for w in Worker._instances if w.status in RUNNING)
         except StopIteration:
             raise ValueError("No workers found")