diff --git a/fastapi/applications.py b/fastapi/applications.py
index 05c7bd2be..348884dc5 100644
--- a/fastapi/applications.py
+++ b/fastapi/applications.py
@@ -512,6 +512,14 @@ class FastAPI(Starlette):
                 Read more in the [FastAPI docs for `lifespan`](https://fastapi.tiangolo.com/advanced/events/).
                 """
             ),
+            deprecated(
+                """
+                **Attention**: This parameter is deprecated.
+
+                The recommended approach for managing startup and shutdown
+                is by utilizing the lifespan parameter within the FastAPI app.
+                """
+            ),
         ] = None,
         on_shutdown: Annotated[
             Optional[Sequence[Callable[[], Any]]],
@@ -525,6 +533,14 @@ class FastAPI(Starlette):
                 [FastAPI docs for `lifespan`](https://fastapi.tiangolo.com/advanced/events/).
                 """
             ),
+            deprecated(
+                """
+                **Attention**: This parameter is deprecated.
+
+                The recommended approach for managing startup and shutdown
+                is by utilizing the lifespan parameter within the FastAPI app.
+                """
+            ),
         ] = None,
         lifespan: Annotated[
             Optional[Lifespan[AppType]],
diff --git a/fastapi/routing.py b/fastapi/routing.py
index 54c75a027..dbd06cab9 100644
--- a/fastapi/routing.py
+++ b/fastapi/routing.py
@@ -764,6 +764,14 @@ class APIRouter(routing.Router):
                 Read more in the [FastAPI docs for `lifespan`](https://fastapi.tiangolo.com/advanced/events/).
                 """
             ),
+            deprecated(
+                """
+                **Attention**: This parameter is deprecated.
+
+                The recommended approach for managing startup and shutdown
+                is by utilizing the lifespan parameter within the APIRouter.
+                """
+            ),
         ] = None,
         on_shutdown: Annotated[
             Optional[Sequence[Callable[[], Any]]],
@@ -777,6 +785,14 @@ class APIRouter(routing.Router):
                 [FastAPI docs for `lifespan`](https://fastapi.tiangolo.com/advanced/events/).
                 """
             ),
+            deprecated(
+                """
+                **Attention**: This parameter is deprecated.
+
+                The recommended approach for managing startup and shutdown
+                is by utilizing the lifespan parameter within the APIRouter.
+                """
+            ),
         ] = None,
         # the generic to Lifespan[AppType] is the type of the top level application
         # which the router cannot know statically, so we use typing.Any