diff --git a/distributed/queues.py b/distributed/queues.py
index d022b010e1c..e368d329d03 100644
--- a/distributed/queues.py
+++ b/distributed/queues.py
@@ -7,7 +7,7 @@
 
 from .client import Future, Client
 from .utils import sync, thread_state
-from .worker import get_client
+from .worker import get_client, get_worker
 from .utils import parse_timedelta
 
 logger = logging.getLogger(__name__)
@@ -150,8 +150,8 @@ class Queue:
         Name used by other clients and the scheduler to identify the queue. If
         not given, a random name will be generated.
     client: Client (optional)
-        Client used for communication with the scheduler. Defaults to the
-        value of ``Client.current()``.
+        Client used for communication with the scheduler.
+        If not given, the default global client will be used.
     maxsize: int (optional)
         Number of items allowed in the queue. If 0 (the default), the queue
         size is unbounded.
@@ -170,7 +170,11 @@ class Queue:
     """
 
     def __init__(self, name=None, client=None, maxsize=0):
-        self.client = client or Client.current()
+        try:
+            self.client = client or Client.current()
+        except ValueError:
+            # Initialise new client
+            self.client = get_worker().client
         self.name = name or "queue-" + uuid.uuid4().hex
         self._event_started = asyncio.Event()
         if self.client.asynchronous or getattr(
diff --git a/distributed/tests/test_queues.py b/distributed/tests/test_queues.py
index 94d80c9dbcf..8f400498854 100644
--- a/distributed/tests/test_queues.py
+++ b/distributed/tests/test_queues.py
@@ -6,7 +6,7 @@
 
 from distributed import Client, Queue, Nanny, worker_client, wait, TimeoutError
 from distributed.metrics import time
-from distributed.utils_test import gen_cluster, inc, div
+from distributed.utils_test import gen_cluster, inc, div, popen
 from distributed.utils_test import client, cluster_fixture, loop  # noqa: F401
 
 
@@ -276,3 +276,22 @@ def get():
     res = c.submit(get)
 
     await c.gather([res, fut])
+
+
+def test_queue_in_task(loop):
+    # Ensure that we can create a Queue inside a task on a
+    # worker in a separate Python process than the client
+    with popen(["dask-scheduler", "--no-dashboard"]):
+        with popen(["dask-worker", "127.0.0.1:8786"]):
+            with Client("tcp://127.0.0.1:8786", loop=loop) as c:
+                c.wait_for_workers(1)
+
+                x = Queue("x")
+                x.put(123)
+
+                def foo():
+                    y = Queue("x")
+                    return y.get()
+
+                result = c.submit(foo).result()
+                assert result == 123
diff --git a/distributed/tests/test_variable.py b/distributed/tests/test_variable.py
index 5d9ece6ee54..37b3c756be7 100644
--- a/distributed/tests/test_variable.py
+++ b/distributed/tests/test_variable.py
@@ -10,9 +10,8 @@
 from distributed import Client, Variable, worker_client, Nanny, wait, TimeoutError
 from distributed.metrics import time
 from distributed.compatibility import WINDOWS
-from distributed.utils_test import gen_cluster, inc, div
+from distributed.utils_test import gen_cluster, inc, div, captured_logger, popen
 from distributed.utils_test import client, cluster_fixture, loop  # noqa: F401
-from distributed.utils_test import captured_logger
 
 
 @gen_cluster(client=True)
@@ -40,6 +39,25 @@ async def test_variable(c, s, a, b):
         assert time() < start + 5
 
 
+def test_variable_in_task(loop):
+    # Ensure that we can create a Variable inside a task on a
+    # worker in a separate Python process than the client
+    with popen(["dask-scheduler", "--no-dashboard"]):
+        with popen(["dask-worker", "127.0.0.1:8786"]):
+            with Client("tcp://127.0.0.1:8786", loop=loop) as c:
+                c.wait_for_workers(1)
+
+                x = Variable("x")
+                x.set(123)
+
+                def foo():
+                    y = Variable("x")
+                    return y.get()
+
+                result = c.submit(foo).result()
+                assert result == 123
+
+
 @gen_cluster(client=True)
 async def test_delete_unset_variable(c, s, a, b):
     x = Variable()
diff --git a/distributed/variable.py b/distributed/variable.py
index 19fbd2bb031..c3fdc94d0d7 100644
--- a/distributed/variable.py
+++ b/distributed/variable.py
@@ -9,13 +9,13 @@
 from dask.utils import stringify
 from .client import Future, Client
 from .utils import log_errors, TimeoutError, parse_timedelta
-from .worker import get_client
+from .worker import get_client, get_worker
 
 logger = logging.getLogger(__name__)
 
 
 class VariableExtension:
-    """An extension for the scheduler to manage queues
+    """An extension for the scheduler to manage Variables
 
     This adds the following routes to the scheduler
 
@@ -145,8 +145,8 @@ class Variable:
         Name used by other clients and the scheduler to identify the variable.
         If not given, a random name will be generated.
     client: Client (optional)
-        Client used for communication with the scheduler. Defaults to the
-        value of ``Client.current()``.
+        Client used for communication with the scheduler.
+        If not given, the default global client will be used.
 
     Examples
     --------
@@ -165,7 +165,11 @@ class Variable:
     """
 
     def __init__(self, name=None, client=None, maxsize=0):
-        self.client = client or Client.current()
+        try:
+            self.client = client or Client.current()
+        except ValueError:
+            # Initialise new client
+            self.client = get_worker().client
         self.name = name or "variable-" + uuid.uuid4().hex
 
     async def _set(self, value):