diff --git a/.vscode/cspell.json b/.vscode/cspell.json
index 23a0eb6ebb63..4ef02106acf0 100644
--- a/.vscode/cspell.json
+++ b/.vscode/cspell.json
@@ -156,6 +156,8 @@
     "spinup",
     "cibuildwheel",
     "aoai",
+    "aiotest",
+    "aiotests",
     "pyprojects",
     "certifi",
     "cffi",
diff --git a/sdk/core/azure-core/CHANGELOG.md b/sdk/core/azure-core/CHANGELOG.md
index b8f7e4dbe1ff..83eaa3394cbe 100644
--- a/sdk/core/azure-core/CHANGELOG.md
+++ b/sdk/core/azure-core/CHANGELOG.md
@@ -1,15 +1,15 @@
 # Release History
 
-## 1.35.0 (Unreleased)
+## 1.35.0 (2025-06-05)
 
 ### Features Added
 
 - Added a `start_time` keyword argument to the `start_span` and `start_as_current_span` methods in the `OpenTelemetryTracer` class. This allows users to specify a custom start time for created spans. #41106
 
-### Breaking Changes
-
 ### Bugs Fixed
 
+- Reduce risk of hanging while closing aiohttp transport if server does not follow best practices. Fix #41363
+
 ### Other Changes
 
 - A timeout error when using the `aiohttp` transport (the default for async SDKs) will now be raised as a `azure.core.exceptions.ServiceResponseTimeoutError`, a subtype of the previously raised `ServiceResponseError`.
diff --git a/sdk/core/azure-core/azure/core/pipeline/transport/_aiohttp.py b/sdk/core/azure-core/azure/core/pipeline/transport/_aiohttp.py
index 922f08d1995f..4a7ab3054678 100644
--- a/sdk/core/azure-core/azure/core/pipeline/transport/_aiohttp.py
+++ b/sdk/core/azure-core/azure/core/pipeline/transport/_aiohttp.py
@@ -168,7 +168,10 @@ async def open(self):
     async def close(self):
         """Closes the connection."""
         if self._session_owner and self.session:
-            await self.session.close()
+            try:
+                await asyncio.wait_for(self.session.close(), timeout=0.0001)  # close immediately
+            except (asyncio.TimeoutError, TimeoutError):
+                pass
             self.session = None
 
     def _build_ssl_config(self, cert, verify):
diff --git a/sdk/core/azure-core/tests/async_tests/test_aiohttp_blob_download_live_async.py b/sdk/core/azure-core/tests/async_tests/test_aiohttp_blob_download_live_async.py
new file mode 100644
index 000000000000..028134a927fd
--- /dev/null
+++ b/sdk/core/azure-core/tests/async_tests/test_aiohttp_blob_download_live_async.py
@@ -0,0 +1,33 @@
+# -------------------------------------------------------------------------
+# Copyright (c) Microsoft Corporation. All rights reserved.
+# Licensed under the MIT License. See LICENSE.txt in the project root for
+# license information.
+# -------------------------------------------------------------------------
+import sys
+import pytest
+import logging
+from azure.storage.blob.aio import BlobServiceClient
+
+
+@pytest.mark.skipif(
+    sys.version_info < (3, 11), reason="ssl_shutdown_timeout in aiohttp only takes effect on Python 3.11+"
+)
+@pytest.mark.live_test_only
+@pytest.mark.asyncio
+async def test_download_blob_aiohttp(caplog):
+    logger = logging.getLogger(__name__)
+    AZURE_STORAGE_CONTAINER_NAME = "aiotests"
+    account_url = "https://aiotests.blob.core.windows.net"
+    blob_service_client = BlobServiceClient(account_url=account_url)
+    read_path = "aiotest.txt"
+
+    with caplog.at_level(logging.INFO):
+        async with blob_service_client:
+            blob_client = blob_service_client.get_blob_client(container=AZURE_STORAGE_CONTAINER_NAME, blob=read_path)
+
+            async with blob_client:
+                stream = await blob_client.download_blob()
+                data = await stream.readall()
+                logger.info(f"Blob size: {len(data)}")
+
+        assert all("Error" not in message for message in caplog.messages)
diff --git a/sdk/core/azure-core/tests/async_tests/test_aiohttp_close_timeout.py b/sdk/core/azure-core/tests/async_tests/test_aiohttp_close_timeout.py
new file mode 100644
index 000000000000..74b456ec51d5
--- /dev/null
+++ b/sdk/core/azure-core/tests/async_tests/test_aiohttp_close_timeout.py
@@ -0,0 +1,58 @@
+# -------------------------------------------------------------------------
+# Copyright (c) Microsoft Corporation. All rights reserved.
+# Licensed under the MIT License. See LICENSE.txt in the project root for
+# license information.
+# -------------------------------------------------------------------------
+import asyncio
+import time
+import pytest
+import types
+from unittest.mock import Mock, patch
+
+from azure.core.pipeline.transport import (
+    AioHttpTransport,
+)
+
+
+class MockClientSession:
+    """Mock aiohttp.ClientSession with a close method that sleeps for 30 seconds."""
+
+    def __init__(self):
+        self._closed = False
+
+    async def __aenter__(self):
+        return self
+
+    async def __aexit__(self, *args, **kwargs):
+        pass
+
+    async def close(self):
+        """Simulate a slow closing session."""
+        self._closed = True
+        await asyncio.sleep(30)  # Simulate a very slow close operation
+
+    def request(self, *args, **kwargs):
+        return asyncio.Future()  # Just needs to be awaitable
+
+
+@pytest.mark.asyncio
+async def test_aiohttp_transport_close_timeout():
+    """Test that AioHttpTransport.close() returns within 1 second even if session.close() takes 30 seconds."""
+
+    # Create transport with our mock session
+    mock_session = MockClientSession()
+    transport = AioHttpTransport(session=mock_session, session_owner=True)
+
+    # Open the transport to initialize the session
+    await transport.open()
+
+    # Time the close operation
+    start_time = time.time()
+    await transport.close()
+    end_time = time.time()
+
+    # Verify close returned in a reasonable time (should be around 0.0001 seconds due to timeout)
+    assert end_time - start_time < 0.1, f"Transport close took {end_time - start_time} seconds, should be < 0.1 seconds"
+
+    # Ensure transport's session was set to None
+    assert transport.session is None