diff --git a/starlette/middleware/base.py b/starlette/middleware/base.py
index 8f2f02ea06..c9880206e2 100644
--- a/starlette/middleware/base.py
+++ b/starlette/middleware/base.py
@@ -2,8 +2,9 @@
 
 import anyio
 
+from starlette.background import BackgroundTask
 from starlette.requests import Request
-from starlette.responses import Response, StreamingResponse
+from starlette.responses import ContentStream, Response, StreamingResponse
 from starlette.types import ASGIApp, Message, Receive, Scope, Send
 
 RequestResponseEndpoint = typing.Callable[[Request], typing.Awaitable[Response]]
@@ -75,6 +76,9 @@ async def coro() -> None:
 
             try:
                 message = await recv_stream.receive()
+                info = message.get("info", None)
+                if message["type"] == "http.response.debug" and info is not None:
+                    message = await recv_stream.receive()
             except anyio.EndOfStream:
                 if app_exc is not None:
                     raise app_exc
@@ -93,8 +97,8 @@ async def body_stream() -> typing.AsyncGenerator[bytes, None]:
                 if app_exc is not None:
                     raise app_exc
 
-            response = StreamingResponse(
-                status_code=message["status"], content=body_stream()
+            response = _StreamingResponse(
+                status_code=message["status"], content=body_stream(), info=info
             )
             response.raw_headers = message["headers"]
             return response
@@ -109,3 +113,22 @@ async def dispatch(
         self, request: Request, call_next: RequestResponseEndpoint
     ) -> Response:
         raise NotImplementedError()  # pragma: no cover
+
+
+class _StreamingResponse(StreamingResponse):
+    def __init__(
+        self,
+        content: ContentStream,
+        status_code: int = 200,
+        headers: typing.Optional[typing.Mapping[str, str]] = None,
+        media_type: typing.Optional[str] = None,
+        background: typing.Optional[BackgroundTask] = None,
+        info: typing.Optional[typing.Mapping[str, typing.Any]] = None,
+    ) -> None:
+        self._info = info
+        super().__init__(content, status_code, headers, media_type, background)
+
+    async def stream_response(self, send: Send) -> None:
+        if self._info:
+            await send({"type": "http.response.debug", "info": self._info})
+        return await super().stream_response(send)
diff --git a/starlette/templating.py b/starlette/templating.py
index 7c46a65a0b..ecea4f3a3d 100644
--- a/starlette/templating.py
+++ b/starlette/templating.py
@@ -42,12 +42,14 @@ def __init__(
     async def __call__(self, scope: Scope, receive: Receive, send: Send) -> None:
         request = self.context.get("request", {})
         extensions = request.get("extensions", {})
-        if "http.response.template" in extensions:
+        if "http.response.debug" in extensions:
             await send(
                 {
-                    "type": "http.response.template",
-                    "template": self.template,
-                    "context": self.context,
+                    "type": "http.response.debug",
+                    "info": {
+                        "template": self.template,
+                        "context": self.context,
+                    },
                 }
             )
         await super().__call__(scope, receive, send)
diff --git a/starlette/testclient.py b/starlette/testclient.py
index 48afcb6734..549fa76219 100644
--- a/starlette/testclient.py
+++ b/starlette/testclient.py
@@ -259,7 +259,7 @@ def handle_request(self, request: httpx.Request) -> httpx.Response:
             "headers": headers,
             "client": ["testclient", 50000],
             "server": [host, port],
-            "extensions": {"http.response.template": {}},
+            "extensions": {"http.response.debug": {}},
         }
 
         request_complete = False
@@ -324,9 +324,9 @@ async def send(message: Message) -> None:
                 if not more_body:
                     raw_kwargs["stream"].seek(0)
                     response_complete.set()
-            elif message["type"] == "http.response.template":
-                template = message["template"]
-                context = message["context"]
+            elif message["type"] == "http.response.debug":
+                template = message["info"]["template"]
+                context = message["info"]["context"]
 
         try:
             with self.portal_factory() as portal:
diff --git a/tests/test_templates.py b/tests/test_templates.py
index 0bf4bce075..2d918f7d0c 100644
--- a/tests/test_templates.py
+++ b/tests/test_templates.py
@@ -3,6 +3,8 @@
 import pytest
 
 from starlette.applications import Starlette
+from starlette.middleware import Middleware
+from starlette.middleware.base import BaseHTTPMiddleware
 from starlette.routing import Route
 from starlette.templating import Jinja2Templates
 
@@ -60,3 +62,29 @@ def hello_world_processor(request):
     assert response.text == "<html>Hello World</html>"
     assert response.template.name == "index.html"
     assert set(response.context.keys()) == {"request", "username"}
+
+
+def test_template_with_middleware(tmpdir, test_client_factory):
+    path = os.path.join(tmpdir, "index.html")
+    with open(path, "w") as file:
+        file.write("<html>Hello, <a href='{{ url_for('homepage') }}'>world</a></html>")
+
+    async def homepage(request):
+        return templates.TemplateResponse("index.html", {"request": request})
+
+    class CustomMiddleware(BaseHTTPMiddleware):
+        async def dispatch(self, request, call_next):
+            return await call_next(request)
+
+    app = Starlette(
+        debug=True,
+        routes=[Route("/", endpoint=homepage)],
+        middleware=[Middleware(CustomMiddleware)],
+    )
+    templates = Jinja2Templates(directory=str(tmpdir))
+
+    client = test_client_factory(app)
+    response = client.get("/")
+    assert response.text == "<html>Hello, <a href='http://testserver/'>world</a></html>"
+    assert response.template.name == "index.html"
+    assert set(response.context.keys()) == {"request"}