diff --git a/src/websockets/asyncio/__init__.py b/src/websockets/asyncio/__init__.py
new file mode 100644
index 00000000..e69de29b
diff --git a/src/websockets/legacy/async_timeout.py b/src/websockets/asyncio/async_timeout.py
similarity index 100%
rename from src/websockets/legacy/async_timeout.py
rename to src/websockets/asyncio/async_timeout.py
diff --git a/src/websockets/legacy/compatibility.py b/src/websockets/asyncio/compatibility.py
similarity index 100%
rename from src/websockets/legacy/compatibility.py
rename to src/websockets/asyncio/compatibility.py
diff --git a/src/websockets/legacy/client.py b/src/websockets/legacy/client.py
index 48622523..586ba528 100644
--- a/src/websockets/legacy/client.py
+++ b/src/websockets/legacy/client.py
@@ -20,6 +20,7 @@
     cast,
 )
 
+from ..asyncio.compatibility import asyncio_timeout
 from ..datastructures import Headers, HeadersLike
 from ..exceptions import (
     InvalidHandshake,
@@ -44,7 +45,6 @@
 from ..http import USER_AGENT
 from ..typing import ExtensionHeader, LoggerLike, Origin, Subprotocol
 from ..uri import WebSocketURI, parse_uri
-from .compatibility import asyncio_timeout
 from .handshake import build_request, check_response
 from .http import read_response
 from .protocol import WebSocketCommonProtocol
diff --git a/src/websockets/legacy/protocol.py b/src/websockets/legacy/protocol.py
index 0422c10d..9d2fe6e7 100644
--- a/src/websockets/legacy/protocol.py
+++ b/src/websockets/legacy/protocol.py
@@ -28,6 +28,7 @@
     cast,
 )
 
+from ..asyncio.compatibility import asyncio_timeout
 from ..datastructures import Headers
 from ..exceptions import (
     ConnectionClosed,
@@ -53,7 +54,6 @@
 )
 from ..protocol import State
 from ..typing import Data, LoggerLike, Subprotocol
-from .compatibility import asyncio_timeout
 from .framing import Frame
 
 
diff --git a/src/websockets/legacy/server.py b/src/websockets/legacy/server.py
index a17c5232..23322050 100644
--- a/src/websockets/legacy/server.py
+++ b/src/websockets/legacy/server.py
@@ -25,6 +25,7 @@
     cast,
 )
 
+from ..asyncio.compatibility import asyncio_timeout
 from ..datastructures import Headers, HeadersLike, MultipleValuesError
 from ..exceptions import (
     AbortHandshake,
@@ -46,7 +47,6 @@
 from ..http import USER_AGENT
 from ..protocol import State
 from ..typing import ExtensionHeader, LoggerLike, Origin, Subprotocol
-from .compatibility import asyncio_timeout
 from .handshake import build_response, check_request
 from .http import read_request
 from .protocol import WebSocketCommonProtocol
diff --git a/tests/legacy/test_client_server.py b/tests/legacy/test_client_server.py
index 133af053..0c96e821 100644
--- a/tests/legacy/test_client_server.py
+++ b/tests/legacy/test_client_server.py
@@ -14,6 +14,7 @@
 import urllib.request
 import warnings
 
+from websockets.asyncio.compatibility import asyncio_timeout
 from websockets.datastructures import Headers
 from websockets.exceptions import (
     ConnectionClosed,
@@ -29,7 +30,6 @@
 )
 from websockets.http import USER_AGENT
 from websockets.legacy.client import *
-from websockets.legacy.compatibility import asyncio_timeout
 from websockets.legacy.handshake import build_response
 from websockets.legacy.http import read_response
 from websockets.legacy.server import *