diff --git a/llvm/lib/Transforms/IPO/FunctionSpecialization.cpp b/llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
--- a/llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
+++ b/llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
@@ -677,8 +677,12 @@
       auto *V = CS.getArgOperand(A->getArgNo());
       if (isa<PoisonValue>(V))
         return false;
-      if (isa<ConstantExpr>(V))
-        return false;
+
+      // For now, constant expressions are fine but only if they are function
+      // calls.
+      if (auto *CE =  dyn_cast<ConstantExpr>(V))
+        if (!isa<Function>(CE->getOperand(0)))
+          return false;
 
       // TrackValueOfGlobalVariable only tracks scalar global variables.
       if (auto *GV = dyn_cast<GlobalVariable>(V)) {
diff --git a/llvm/test/Transforms/FunctionSpecialization/function-specialization-constant-expression3.ll b/llvm/test/Transforms/FunctionSpecialization/function-specialization-constant-expression3.ll
new file mode 100644
--- /dev/null
+++ b/llvm/test/Transforms/FunctionSpecialization/function-specialization-constant-expression3.ll
@@ -0,0 +1,29 @@
+; NOTE: Assertions have been autogenerated by utils/update_test_checks.py
+; RUN: opt -function-specialization -force-function-specialization -S < %s | FileCheck %s
+
+define i32 @main() {
+; CHECK-LABEL: @main(
+; CHECK-NEXT:  bb:
+; CHECK-NEXT:    tail call void @wombat.1(i8* undef, i64 undef, i64 undef, i32 (i8*, i8*)* bitcast (i32 ()* @quux to i32 (i8*, i8*)*))
+; CHECK-NEXT:    tail call void @wombat.2(i8* undef, i64 undef, i64 undef, i32 (i8*, i8*)* bitcast (i32 ()* @eggs to i32 (i8*, i8*)*))
+; CHECK-NEXT:    ret i32 undef
+;
+bb:
+  tail call void @wombat(i8* undef, i64 undef, i64 undef, i32 (i8*, i8*)* bitcast (i32 ()* @quux to i32 (i8*, i8*)*))
+  tail call void @wombat(i8* undef, i64 undef, i64 undef, i32 (i8*, i8*)* bitcast (i32 ()* @eggs to i32 (i8*, i8*)*))
+  ret i32 undef
+}
+
+declare i32 @quux()
+declare i32 @eggs()
+
+define internal void @wombat(i8* %arg, i64 %arg1, i64 %arg2, i32 (i8*, i8*)* %arg3) {
+; CHECK-LABEL: @wombat(
+; CHECK-NEXT:  bb4:
+; CHECK-NEXT:    [[TMP:%.*]] = tail call i32 [[ARG3:%.*]](i8* undef, i8* undef)
+; CHECK-NEXT:    ret void
+;
+bb4:
+  %tmp = tail call i32 %arg3(i8* undef, i8* undef)
+  ret void
+}