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
@@ -731,18 +731,17 @@
 
       auto *V = CS.getArgOperand(A->getArgNo());
       if (isa<PoisonValue>(V))
-        return;
+        continue;
 
       // TrackValueOfGlobalVariable only tracks scalar global variables.
       if (auto *GV = dyn_cast<GlobalVariable>(V)) {
         // Check if we want to specialize on the address of non-constant
         // global values.
-        if (!GV->isConstant())
-          if (!SpecializeOnAddresses)
-            return;
+        if (!GV->isConstant() && !SpecializeOnAddresses)
+          continue;
 
         if (!GV->getValueType()->isSingleValueType())
-          return;
+          continue;
       }
 
       if (isa<Constant>(V) && (Solver.getLatticeValueFor(V).isConstant() ||
diff --git a/llvm/test/Transforms/FunctionSpecialization/get-possible-constants.ll b/llvm/test/Transforms/FunctionSpecialization/get-possible-constants.ll
new file mode 100644
--- /dev/null
+++ b/llvm/test/Transforms/FunctionSpecialization/get-possible-constants.ll
@@ -0,0 +1,42 @@
+; RUN: opt -S --passes=function-specialization < %s | FileCheck %s
+define dso_local i32 @p0(i32 noundef %x) {
+entry:
+  %add = add nsw i32 %x, 1
+  ret i32 %add
+}
+
+define dso_local i32 @p1(i32 noundef %x) {
+entry:
+  %sub = add nsw i32 %x, -1
+  ret i32 %sub
+}
+
+define dso_local i32 @g0(i32 noundef %x) {
+entry:
+  %call = tail call fastcc i32 @f(i32 noundef %x, ptr noundef nonnull @p0)
+  ret i32 %call
+}
+
+define internal fastcc i32 @f(i32 noundef %x, ptr nocapture noundef readonly %p) noinline  {
+entry:
+  %call = tail call i32 %p(i32 noundef %x)
+  %add = add nsw i32 %call, %x
+  ret i32 %add
+}
+
+define dso_local i32 @g1(i32 noundef %x) {
+entry:
+  %call = tail call fastcc i32 @f(i32 noundef %x, ptr noundef nonnull @p1)
+  ret i32 %call
+}
+
+define dso_local i32 @g2(i32 noundef %x) {
+entry:
+  %call = tail call fastcc i32 @f(i32 noundef %x, ptr poison)
+  ret i32 %call
+}
+
+; Check that a single argument, that cannot be used for specialisation, does not
+; prevent specialisation based on other arguments.
+; CHECK: @f.1
+; CHECK: @f.2