diff --git a/fastapi/applications.py b/fastapi/applications.py
index 3021d7593..582e9765c 100644
--- a/fastapi/applications.py
+++ b/fastapi/applications.py
@@ -515,6 +515,14 @@ 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]]],
@@ -528,6 +536,14 @@ 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 54d53bbbf..2082ab0ab 100644
--- a/fastapi/routing.py
+++ b/fastapi/routing.py
@@ -690,6 +690,14 @@ 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]]],
@@ -703,6 +711,14 @@ 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