diff --git a/mypy/checkexpr.py b/mypy/checkexpr.py
index e8a2e501a452..717bf737ac46 100644
--- a/mypy/checkexpr.py
+++ b/mypy/checkexpr.py
@@ -2820,6 +2820,7 @@ def infer_overload_return_type(
         matches: list[CallableType] = []
         return_types: list[Type] = []
         inferred_types: list[Type] = []
+        self_contains_any = has_any_type(object_type) if object_type is not None else False
         args_contain_any = any(map(has_any_type, arg_types))
         type_maps: list[dict[Expression, Type]] = []
 
@@ -2840,7 +2841,7 @@ def infer_overload_return_type(
             if is_match:
                 # Return early if possible; otherwise record info, so we can
                 # check for ambiguity due to 'Any' below.
-                if not args_contain_any:
+                if not args_contain_any and not self_contains_any:
                     self.chk.store_types(m)
                     return ret_type, infer_type
                 p_infer_type = get_proper_type(infer_type)
@@ -2856,7 +2857,14 @@ def infer_overload_return_type(
 
         if not matches:
             return None
-        elif any_causes_overload_ambiguity(matches, return_types, arg_types, arg_kinds, arg_names):
+        elif any_causes_overload_ambiguity(
+            matches,
+            return_types,
+            arg_types,
+            arg_kinds,
+            arg_names,
+            self_contains_any=self_contains_any,
+        ):
             # An argument of type or containing the type 'Any' caused ambiguity.
             # We try returning a precise type if we can. If not, we give up and just return 'Any'.
             if all_same_types(return_types):
@@ -6452,6 +6460,8 @@ def any_causes_overload_ambiguity(
     arg_types: list[Type],
     arg_kinds: list[ArgKind],
     arg_names: Sequence[str | None] | None,
+    *,
+    self_contains_any: bool = False,
 ) -> bool:
     """May an argument containing 'Any' cause ambiguous result type on call to overloaded function?
 
@@ -6501,7 +6511,7 @@ def any_causes_overload_ambiguity(
             if not all_same_types(matching_formals) and not all_same_types(matching_returns):
                 # Any maps to multiple different types, and the return types of these items differ.
                 return True
-    return False
+    return self_contains_any
 
 
 def all_same_types(types: list[Type]) -> bool:
diff --git a/test-data/unit/check-overloading.test b/test-data/unit/check-overloading.test
index 7bca5cc7b508..6d4d83b72897 100644
--- a/test-data/unit/check-overloading.test
+++ b/test-data/unit/check-overloading.test
@@ -6693,3 +6693,65 @@ class B:
     def f(self, *args, **kwargs):
         pass
 [builtins fixtures/tuple.pyi]
+
+[case testOverloadSelfArgWithMultipleMatches]
+# https://github.com/python/mypy/issues/11347
+from typing import Generic, TypeVar, overload, Any
+
+T = TypeVar('T')
+
+class Some(Generic[T]):
+    @overload
+    def method(self: Some[int]) -> str: ...
+    @overload
+    def method(self: Some[str]) -> float: ...
+    def method(self): ...
+
+s1: Some[int]
+s2: Some[str]
+s3: Some[Any]
+reveal_type(s1.method())  # N: Revealed type is "builtins.str"
+reveal_type(s2.method())  # N: Revealed type is "builtins.float"
+reveal_type(s3.method()) # N: Revealed type is "Any"
+[builtins fixtures/dict.pyi]
+
+[case testOverloadSelfArgWithOtherSameArgAndMultipleMatches]
+from typing import Generic, TypeVar, overload, Any
+
+T = TypeVar('T')
+
+class Some(Generic[T]):
+    @overload
+    def method(self: Some[int], other: int) -> str: ...
+    @overload
+    def method(self: Some[str], other: int) -> float: ...
+    def method(self): ...
+
+# was ok
+s1: Some[int]
+s2: Some[str]
+s3: Some[Any]
+reveal_type(s1.method(1))  # N: Revealed type is "builtins.str"
+reveal_type(s2.method(1))  # N: Revealed type is "builtins.float"
+reveal_type(s3.method(1)) # N: Revealed type is "Any"
+[builtins fixtures/dict.pyi]
+
+[case testOverloadSelfArgWithOtherDifferentArgAndMultipleMatches]
+from typing import Generic, TypeVar, overload, Any
+
+T = TypeVar('T')
+
+class Some(Generic[T]):
+    @overload
+    def method(self: Some[int], other: int) -> str: ...
+    @overload
+    def method(self: Some[str], other: str) -> float: ...
+    def method(self): ...
+
+s1: Some[int]
+s2: Some[str]
+s3: Some[Any]
+reveal_type(s1.method(1))  # N: Revealed type is "builtins.str"
+reveal_type(s2.method('a'))  # N: Revealed type is "builtins.float"
+reveal_type(s3.method(1)) # N: Revealed type is "builtins.str"
+[builtins fixtures/dict.pyi]