diff --git a/repro.py b/repro.py
new file mode 100644
index 00000000..8fda0e9c
--- /dev/null
+++ b/repro.py
@@ -0,0 +1,84 @@
+import asyncio
+import logging
+import json
+from websockets.asyncio.client import connect
+
+# import debugpy
+
+# # Allow VS Code to attach
+# debugpy.listen(("0.0.0.0", 5678))  # Use the port you've specified
+# print("Waiting for debugger to attach...")
+# debugpy.wait_for_client()
+
+logging.getLogger(__name__)
+logging.basicConfig(
+    format="%(asctime)s %(module)s:%(lineno)d %(levelname)8s | %(message)s",
+    datefmt="%Y/%m/%d %H:%M:%S",
+    level=logging.DEBUG,
+)
+
+class MyClient:
+
+    def __init__(self):
+        self.keep_alive = True
+
+    async def run(self):
+
+        async with connect(
+            f"wss://ws.kraken.com/v2",
+            ping_interval=30,
+            # max_queue=None,  # having this enabled doesn't cause problems
+        ) as socket:
+            await socket.send(
+                json.dumps(
+                    {
+                        "method": "subscribe",
+                        "params": {
+                            "channel": "book",
+                            "symbol": [
+                                "BTC/USD",
+                                "DOT/USD",
+                                "ETH/USD",
+                                "MATIC/USD",
+                                "BTC/EUR",
+                                "DOT/EUR",
+                                "ETH/EUR",
+                                "XLM/USD",
+                                "XLM/EUR",
+                            ],
+                            "depth": 100
+                        },
+                    }
+                )
+            )
+
+            while self.keep_alive:
+                try:
+                    _message = await asyncio.wait_for(socket.recv(), timeout=10)
+                except TimeoutError:
+                    pass
+                except asyncio.CancelledError:
+                    self.keep_alive = False
+                else:
+                    try:
+                        message = json.loads(_message)
+                    except ValueError:
+                        pass
+
+    async def __aenter__(self):
+        self.task: asyncio.Task = asyncio.create_task(self.run())
+        return self
+
+    async def __aexit__(self, *args, **kwargs):
+        self.keep_alive = False
+        if hasattr(self, "task") and not self.task.done():
+            await self.task
+
+
+async def main():
+    async with MyClient():
+        await asyncio.sleep(3)
+
+
+if __name__ == "__main__":
+    asyncio.run(main())
diff --git a/src/websockets/asyncio/connection.py b/src/websockets/asyncio/connection.py
index e5c350fe..c947f5a4 100644
--- a/src/websockets/asyncio/connection.py
+++ b/src/websockets/asyncio/connection.py
@@ -913,6 +913,7 @@ async def send_context(
         if wait_for_close:
             try:
                 async with asyncio_timeout_at(self.close_deadline):
+                    self.recv_messages.prepare_close()
                     await asyncio.shield(self.connection_lost_waiter)
             except TimeoutError:
                 # There's no risk to overwrite another error because
diff --git a/src/websockets/asyncio/messages.py b/src/websockets/asyncio/messages.py
index e6d1d31c..936adf63 100644
--- a/src/websockets/asyncio/messages.py
+++ b/src/websockets/asyncio/messages.py
@@ -113,6 +113,9 @@ def __init__(  # pragma: no cover
         # This flag prevents concurrent calls to get() by user code.
         self.get_in_progress = False
 
+        # This flag marks a soon end of the connection.
+        self.closing = False
+
         # This flag marks the end of the connection.
         self.closed = False
 
@@ -255,7 +258,9 @@ def put(self, frame: Frame) -> None:
             raise EOFError("stream of frames ended")
 
         self.frames.put(frame)
-        self.maybe_pause()
+
+        if not self.closing:
+            self.maybe_pause()
 
     def maybe_pause(self) -> None:
         """Pause the writer if queue is above the high water mark."""
@@ -279,11 +284,22 @@ def maybe_resume(self) -> None:
             self.paused = False
             self.resume()
 
+    def prepare_close(self) -> None:
+        """
+        Prepare to close by ensuring that no more messages will be processed.
+        """
+        self.closing = True
+
+        # Resuming the writer to avoid deadlocks
+        if self.paused:
+            self.paused = False
+            self.resume()
+
     def close(self) -> None:
         """
         End the stream of frames.
 
-        Callling :meth:`close` concurrently with :meth:`get`, :meth:`get_iter`,
+        Calling :meth:`close` concurrently with :meth:`get`, :meth:`get_iter`,
         or :meth:`put` is safe. They will raise :exc:`EOFError`.
 
         """
diff --git a/src/websockets/sync/messages.py b/src/websockets/sync/messages.py
index 98490797..7d21a4fa 100644
--- a/src/websockets/sync/messages.py
+++ b/src/websockets/sync/messages.py
@@ -291,7 +291,7 @@ def close(self) -> None:
         """
         End the stream of frames.
 
-        Callling :meth:`close` concurrently with :meth:`get`, :meth:`get_iter`,
+        Calling :meth:`close` concurrently with :meth:`get`, :meth:`get_iter`,
         or :meth:`put` is safe. They will raise :exc:`EOFError`.
 
         """
diff --git a/tests/asyncio/test_connection.py b/tests/asyncio/test_connection.py
index 5a0b61bf..95f4e56a 100644
--- a/tests/asyncio/test_connection.py
+++ b/tests/asyncio/test_connection.py
@@ -806,6 +806,27 @@ async def test_close_preserves_queued_messages(self):
         self.assertEqual(str(exc), "sent 1000 (OK); then received 1000 (OK)")
         self.assertIsNone(exc.__cause__)
 
+    async def test_close_preserves_queued_messages_gt_max_queue(self):
+        """
+        close preserves messages buffered in the assembler, even if they
+        exceed the default buffer size.
+        """
+
+        for _ in range(100):
+            await self.remote_connection.send("😀")
+
+        await self.connection.close()
+
+        for _ in range(100):
+            self.assertEqual(await self.connection.recv(), "😀")
+
+        with self.assertRaises(ConnectionClosedOK) as raised:
+            await self.connection.recv()
+
+        exc = raised.exception
+        self.assertEqual(str(exc), "sent 1000 (OK); then received 1000 (OK)")
+        self.assertIsNone(exc.__cause__)
+
     async def test_close_idempotency(self):
         """close does nothing if the connection is already closed."""
         await self.connection.close()