Index: llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
===================================================================
--- llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
+++ llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
@@ -471,17 +471,17 @@
         }
       }
 
-      // Get the lattice value for the value the call site passes to the
-      // argument. If this value is not constant, move on to the next call
-      // site. Additionally, set the AllConstant flag to false.
-      if (V != A && !Solver.getLatticeValueFor(V).isConstant()) {
-        AllConstant = false;
+      // Add the constant to the set.
+      if (auto *C = dyn_cast<Constant>(V)) {
+        Constants.push_back(C);
         continue;
       }
 
-      // Add the constant to the set.
-      if (auto *C = dyn_cast<Constant>(CS.getArgOperand(A->getArgNo())))
-        Constants.push_back(C);
+      // Get the lattice value for the value the call site passes to the
+      // argument. If this value is not constant, set the AllConstant flag to
+      // false.
+      if (!Solver.getLatticeValueFor(V).isConstant())
+        AllConstant = false;
     }
 
     // If the argument can only take on constant values, AllConstant will be
Index: llvm/test/Transforms/FunctionSpecialization/function-specialization-simple.ll
===================================================================
--- /dev/null
+++ llvm/test/Transforms/FunctionSpecialization/function-specialization-simple.ll
@@ -0,0 +1,44 @@
+; RUN: opt -function-specialization -S < %s | FileCheck %s
+
+; Check that the direct constant parameter could be specialized.
+; CHECK: @foo.1(
+; CHECK: @foo.2(
+
+target datalayout = "e-m:e-i8:8:32-i16:16:32-i64:64-i128:128-n32:64-S128"
+
+declare i32 @getValue()
+declare i1 @getCond()
+
+define internal i32 @foo(i1 %break_cond) {
+entry:
+  br label %loop.entry
+
+loop.entry:
+  br label %loop2.entry
+
+loop2.entry:
+  br label %loop2.body
+
+loop2.body:
+  %value = call i32 @getValue()
+  br i1 %break_cond, label %loop2.end, label %return
+
+loop2.end:
+  %cond.end = call i1 @getCond()
+  br i1 %cond.end, label %loop2.entry, label %loop.end
+
+loop.end:
+  %cond2.end = call i1 @getCond()
+  br i1 %cond2.end, label %loop.entry, label %return
+
+return:
+  ret i32 %value
+}
+
+define dso_local i32 @bar(i32 %x, i32 %y) {
+entry:
+  %retval.1 = call i32 @foo(i1 1)
+  %retval.2 = call i32 @foo(i1 0)
+  %retval = add nsw i32 %retval.1, %retval.2
+  ret i32 %retval
+}
\ No newline at end of file