diff --git a/clang-tools-extra/clang-tidy/utils/DeclRefExprUtils.cpp b/clang-tools-extra/clang-tidy/utils/DeclRefExprUtils.cpp
--- a/clang-tools-extra/clang-tidy/utils/DeclRefExprUtils.cpp
+++ b/clang-tools-extra/clang-tidy/utils/DeclRefExprUtils.cpp
@@ -59,9 +59,13 @@
   extractNodesByIdTo(Matches, "declRef", DeclRefs);
   auto ConstReferenceOrValue =
       qualType(anyOf(referenceType(pointee(qualType(isConstQualified()))),
-                     unless(anyOf(referenceType(), pointerType()))));
+                     unless(anyOf(referenceType(), pointerType(),
+                                  substTemplateTypeParmType()))));
+  auto ConstReferenceOrValueOrReplaced = qualType(anyOf(
+      ConstReferenceOrValue,
+      substTemplateTypeParmType(hasReplacementType(ConstReferenceOrValue))));
   auto UsedAsConstRefOrValueArg = forEachArgumentWithParam(
-      DeclRefToVar, parmVarDecl(hasType(ConstReferenceOrValue)));
+      DeclRefToVar, parmVarDecl(hasType(ConstReferenceOrValueOrReplaced)));
   Matches = match(findAll(callExpr(UsedAsConstRefOrValueArg)), Stmt, Context);
   extractNodesByIdTo(Matches, "declRef", DeclRefs);
   Matches =
diff --git a/clang-tools-extra/test/clang-tidy/checkers/performance-unnecessary-copy-initialization.cpp b/clang-tools-extra/test/clang-tidy/checkers/performance-unnecessary-copy-initialization.cpp
--- a/clang-tools-extra/test/clang-tidy/checkers/performance-unnecessary-copy-initialization.cpp
+++ b/clang-tools-extra/test/clang-tidy/checkers/performance-unnecessary-copy-initialization.cpp
@@ -411,14 +411,37 @@
 
 template <class>
 class function;
-template <class R, class... Args>
-class function<R(Args...)> {
+template <class R, class... ArgTypes>
+class function<R(ArgTypes...)> {
 public:
   function();
-  function(const function &other);
-  R operator()(Args &&...args) const;
+  function(const function &Other);
+  R operator()(ArgTypes... Args) const;
+};
+
+template <typename>
+struct remove_reference;
+
+template <typename _Tp>
+struct remove_reference {
+  typedef _Tp type;
+};
+
+template <typename _Tp>
+struct remove_reference<_Tp &> {
+  typedef _Tp type;
+};
+
+template <typename _Tp>
+struct remove_reference<_Tp &&> {
+  typedef _Tp type;
 };
 
+template <typename _Tp>
+constexpr _Tp &&forward(typename remove_reference<_Tp>::type &__t) noexcept {
+  return static_cast<_Tp &&>(__t);
+}
+
 } // namespace __1
 } // namespace std
 
@@ -460,3 +483,19 @@
 }
 
 } // namespace fake
+
+void positiveInvokedOnStdFunction(
+    std::function<void(const ExpensiveToCopyType &)> Update,
+    const ExpensiveToCopyType Orig) {
+  auto Copy = Orig.reference();
+  // CHECK-MESSAGES: [[@LINE-1]]:8: warning: the variable 'Copy' is copy-constructed from a const reference
+  // CHECK-FIXES: const auto& Copy = Orig.reference();
+  Update(Copy);
+}
+
+void negativeInvokedOnStdFunction(
+    std::function<void(ExpensiveToCopyType &)> Update,
+    const ExpensiveToCopyType Orig) {
+  auto Copy = Orig.reference();
+  Update(Copy);
+}