Index: clang/lib/AST/Decl.cpp
===================================================================
--- clang/lib/AST/Decl.cpp
+++ clang/lib/AST/Decl.cpp
@@ -391,11 +391,18 @@
   bool considerVisibility =
     shouldConsiderTemplateVisibility(fn, specInfo);
 
-  // Merge information from the template parameters.
   FunctionTemplateDecl *temp = specInfo->getTemplate();
-  LinkageInfo tempLV =
+
+  // Merge information from the template declaration.
+  LinkageInfo tempLV = getLVForDecl(temp, computation);
+  // The linkage of the specialization should be consistent with the
+  // template declaration.
+  LV.setLinkage(tempLV.getLinkage());
+
+  // Merge information from the template parameters.
+  LinkageInfo paramsLV =
     getLVForTemplateParameterList(temp->getTemplateParameters(), computation);
-  LV.mergeMaybeWithVisibility(tempLV, considerVisibility);
+  LV.mergeMaybeWithVisibility(paramsLV, considerVisibility);
 
   // Merge information from the template arguments.
   const TemplateArgumentList &templateArgs = *specInfo->TemplateArguments;
Index: clang/test/Modules/inconsist-export-template.cpp
===================================================================
--- /dev/null
+++ clang/test/Modules/inconsist-export-template.cpp
@@ -0,0 +1,34 @@
+// RUN: %clang_cc1 -std=c++20 %s -fsyntax-only -verify
+// expected-no-diagnostics
+export module m;
+export template <class T>
+void f() {
+
+}
+
+template <>
+void f<int>() {
+
+}
+
+template <class T>
+void f1() {
+
+}
+
+// FIXME: We should reject following specialization,
+// since it tries to export a name which is already introduced.
+export template <>
+void f1<int>() {
+
+}
+
+export template<class T>
+class C {
+
+};
+
+template<>
+class C<int> {
+
+};