diff --git a/clang/lib/Sema/SemaDecl.cpp b/clang/lib/Sema/SemaDecl.cpp
--- a/clang/lib/Sema/SemaDecl.cpp
+++ b/clang/lib/Sema/SemaDecl.cpp
@@ -13090,9 +13090,10 @@
   // C++ [module.import/6] external definitions are not permitted in header
   // units.
   if (getLangOpts().CPlusPlusModules && currentModuleIsHeaderUnit() &&
-      VDecl->isThisDeclarationADefinition() &&
+      !VDecl->isInvalidDecl() && VDecl->isThisDeclarationADefinition() &&
       VDecl->getFormalLinkage() == Linkage::ExternalLinkage &&
-      !VDecl->isInline()) {
+      !VDecl->isInline() && !VDecl->isTemplated() &&
+      !isa<VarTemplateSpecializationDecl>(VDecl)) {
     Diag(VDecl->getLocation(), diag::err_extern_def_in_header_unit);
     VDecl->setInvalidDecl();
   }
@@ -15261,9 +15262,10 @@
   // FIXME: Consider an alternate location for the test where the inlined()
   // state is complete.
   if (getLangOpts().CPlusPlusModules && currentModuleIsHeaderUnit() &&
+      !FD->isInvalidDecl() && !FD->isInlined() &&
+      BodyKind != FnBodyKind::Delete && BodyKind != FnBodyKind::Default &&
       FD->getFormalLinkage() == Linkage::ExternalLinkage &&
-      !FD->isInvalidDecl() && BodyKind != FnBodyKind::Delete &&
-      BodyKind != FnBodyKind::Default && !FD->isInlined()) {
+      !FD->isTemplated() && !FD->isTemplateInstantiation()) {
     assert(FD->isThisDeclarationADefinition());
     Diag(FD->getLocation(), diag::err_extern_def_in_header_unit);
     FD->setInvalidDecl();
diff --git a/clang/test/CXX/module/module.import/p6.cpp b/clang/test/CXX/module/module.import/p6.cpp
--- a/clang/test/CXX/module/module.import/p6.cpp
+++ b/clang/test/CXX/module/module.import/p6.cpp
@@ -22,6 +22,8 @@
 
 int bad_var_definition = 3;  // expected-error {{non-inline external definitions are not permitted in C++ header units}}
 
+/* The cases below should compile without diagnostics.  */
+
 class A {
 public:
     // This is a declaration instead of definition.
@@ -36,3 +38,32 @@
   S(S&);
 };
 S::S(S&) = default;
+
+template <class _X>
+_X tmpl_var_ok_0 = static_cast<_X>(-1);
+
+template <typename _T>
+constexpr _T tmpl_var_ok_1 = static_cast<_T>(42);
+
+inline int a = tmpl_var_ok_1<int>;
+
+template <typename _Tp,
+          template <typename> class _T>
+constexpr int tmpl_var_ok_2 = _T<_Tp>::value ? 42 : 6174 ;
+
+template<class _Ep>
+int tmpl_OK (_Ep) { return 0; }
+
+template <class _T1>
+bool
+operator==(_T1& , _T1& ) { return false; }
+
+constexpr long one_k = 1000L;
+
+template <class ..._Args>
+void* tmpl_fn_ok
+(_Args ...__args) { return nullptr; }
+
+inline int foo (int a) {
+  return tmpl_OK (a);
+}