diff --git a/mypy/checkmember.py b/mypy/checkmember.py
index 59af0d402e14..4316a59281c3 100644
--- a/mypy/checkmember.py
+++ b/mypy/checkmember.py
@@ -24,6 +24,7 @@
     FuncDef,
     IndexExpr,
     MypyFile,
+    NameExpr,
     OverloadedFuncDef,
     SymbolNode,
     SymbolTable,
@@ -608,7 +609,19 @@ def analyze_member_var_access(
         mx.msg.undefined_in_superclass(name, mx.context)
         return AnyType(TypeOfAny.from_error)
     else:
-        return report_missing_attribute(mx.original_type, itype, name, mx)
+        ret = report_missing_attribute(mx.original_type, itype, name, mx)
+        # Avoid paying double jeopardy if we can't find the member due to --no-implicit-reexport
+        if (
+            mx.module_symbol_table is not None
+            and name in mx.module_symbol_table
+            and not mx.module_symbol_table[name].module_public
+        ):
+            v = mx.module_symbol_table[name].node
+            e = NameExpr(name)
+            e.set_line(mx.context)
+            e.node = v
+            return mx.chk.expr_checker.analyze_ref_expr(e, lvalue=mx.is_lvalue)
+        return ret
 
 
 def check_final_member(name: str, info: TypeInfo, msg: MessageBuilder, ctx: Context) -> None:
diff --git a/test-data/unit/check-flags.test b/test-data/unit/check-flags.test
index 96f78d81dd16..06b7cab8391b 100644
--- a/test-data/unit/check-flags.test
+++ b/test-data/unit/check-flags.test
@@ -1611,14 +1611,22 @@ from other_module_2 import a  # E: Module "other_module_2" does not explicitly e
 reveal_type(a)  # N: Revealed type is "builtins.int"
 
 import other_module_2
-# TODO: this should also reveal builtins.int, see #13965
-reveal_type(other_module_2.a)  # E: "object" does not explicitly export attribute "a"  [attr-defined] \
-                               # N: Revealed type is "Any"
+reveal_type(other_module_2.a)  # E: Module "other_module_2" does not explicitly export attribute "a"  [attr-defined] \
+                               # N: Revealed type is "builtins.int"
+
+from other_module_2 import b  # E: Module "other_module_2" does not explicitly export attribute "b"  [attr-defined]
+reveal_type(b)  # N: Revealed type is "def (a: builtins.int) -> builtins.str"
+
+import other_module_2
+reveal_type(other_module_2.b)  # E: Module "other_module_2" does not explicitly export attribute "b"  [attr-defined] \
+                               # N: Revealed type is "def (a: builtins.int) -> builtins.str"
 
 [file other_module_1.py]
 a = 5
+def b(a: int) -> str: ...
 [file other_module_2.py]
-from other_module_1 import a
+from other_module_1 import a, b
+[builtins fixtures/module.pyi]
 
 [case testNoImplicitReexportRespectsAll]
 # flags: --no-implicit-reexport
@@ -1649,11 +1657,15 @@ __all__ = ('b',)
 [case testNoImplicitReexportGetAttr]
 # flags: --no-implicit-reexport --python-version 3.7
 from other_module_2 import a  # E: Module "other_module_2" does not explicitly export attribute "a"
+reveal_type(a)  # N: Revealed type is "builtins.int"
+from other_module_2 import b  # E: Module "other_module_2" does not explicitly export attribute "b"
+reveal_type(b)  # N: Revealed type is "builtins.str"
 [file other_module_1.py]
-from typing import Any
-def __getattr__(name: str) -> Any: ...
+b: str = "asdf"
+def __getattr__(name: str) -> int: ...
 [file other_module_2.py]
-from other_module_1 import a
+from other_module_1 import a, b
+def __getattr__(name: str) -> bytes: ...
 [builtins fixtures/tuple.pyi]
 
 [case textNoImplicitReexportSuggestions]
diff --git a/test-data/unit/check-modules.test b/test-data/unit/check-modules.test
index 94368f6c1113..abbdf4987c46 100644
--- a/test-data/unit/check-modules.test
+++ b/test-data/unit/check-modules.test
@@ -1862,7 +1862,8 @@ import stub
 
 reveal_type(stub.y)  # N: Revealed type is "builtins.int"
 reveal_type(stub.z)  # E: Module "stub" does not explicitly export attribute "z" \
-                     # N: Revealed type is "Any"
+                     # N: Revealed type is "builtins.int"
+
 
 [file stub.pyi]
 from substub import y as y