diff --git a/mypy/typeshed/stdlib/_typeshed/__init__.pyi b/mypy/typeshed/stdlib/_typeshed/__init__.pyi
index 1b54284fe727..3ae2fca1d19d 100644
--- a/mypy/typeshed/stdlib/_typeshed/__init__.pyi
+++ b/mypy/typeshed/stdlib/_typeshed/__init__.pyi
@@ -276,4 +276,5 @@ StrOrLiteralStr = TypeVar("StrOrLiteralStr", LiteralString, str)  # noqa: Y001
 ProfileFunction: TypeAlias = Callable[[FrameType, str, Any], object]
 
 # Objects suitable to be passed to sys.settrace, threading.settrace, and similar
-TraceFunction: TypeAlias = Callable[[FrameType, str, Any], TraceFunction | None]
+# TODO: Ideally this would be a recursive type alias
+TraceFunction: TypeAlias = Callable[[FrameType, str, Any], Callable[[FrameType, str, Any], Any] | None]
diff --git a/mypy/typeshed/stdlib/builtins.pyi b/mypy/typeshed/stdlib/builtins.pyi
index 00eac9e49cf0..5482955eb0ab 100644
--- a/mypy/typeshed/stdlib/builtins.pyi
+++ b/mypy/typeshed/stdlib/builtins.pyi
@@ -1232,13 +1232,19 @@ def iter(__function: Callable[[], _T | None], __sentinel: None) -> Iterator[_T]:
 @overload
 def iter(__function: Callable[[], _T], __sentinel: object) -> Iterator[_T]: ...
 
+# We need recursive types to express the type of the second argument to `isinstance` properly, hence the use of `Any`
 if sys.version_info >= (3, 10):
-    _ClassInfo: TypeAlias = type | types.UnionType | tuple[_ClassInfo, ...]
+    def isinstance(
+        __obj: object, __class_or_tuple: type | types.UnionType | tuple[type | types.UnionType | tuple[Any, ...], ...]
+    ) -> bool: ...
+    def issubclass(
+        __cls: type, __class_or_tuple: type | types.UnionType | tuple[type | types.UnionType | tuple[Any, ...], ...]
+    ) -> bool: ...
+
 else:
-    _ClassInfo: TypeAlias = type | tuple[_ClassInfo, ...]
+    def isinstance(__obj: object, __class_or_tuple: type | tuple[type | tuple[Any, ...], ...]) -> bool: ...
+    def issubclass(__cls: type, __class_or_tuple: type | tuple[type | tuple[Any, ...], ...]) -> bool: ...
 
-def isinstance(__obj: object, __class_or_tuple: _ClassInfo) -> bool: ...
-def issubclass(__cls: type, __class_or_tuple: _ClassInfo) -> bool: ...
 def len(__obj: Sized) -> int: ...
 def license() -> None: ...
 def locals() -> dict[str, Any]: ...