diff --git a/lib/checkexceptionsafety.cpp b/lib/checkexceptionsafety.cpp
index 97d6f2ef095..98113b9654b 100644
--- a/lib/checkexceptionsafety.cpp
+++ b/lib/checkexceptionsafety.cpp
@@ -25,6 +25,7 @@
 #include "symboldatabase.h"
 #include "token.h"
 
+#include <algorithm>
 #include <list>
 #include <set>
 #include <utility>
@@ -239,6 +240,13 @@ void CheckExceptionSafety::catchExceptionByValueError(const Token *tok)
                 "as a (const) reference which is usually recommended in C++.", CWE398, Certainty::normal);
 }
 
+static bool isPossibleBool(const Token *tok, bool b)
+{
+    const std::list<ValueFlow::Value> &values = tok->values();
+    return std::any_of(values.cbegin(), values.cend(), [&](const ValueFlow::Value &value) {
+        return value.isIntValue() && value.isPossible() && value.intvalue == static_cast<int>(b);
+    });
+}
 
 static const Token * functionThrowsRecursive(const Function * function, std::set<const Function *> & recursive)
 {
@@ -251,6 +259,19 @@ static const Token * functionThrowsRecursive(const Function * function, std::set
 
     for (const Token *tok = function->functionScope->bodyStart->next();
          tok != function->functionScope->bodyEnd; tok = tok->next()) {
+        if (Token::simpleMatch(tok, "if")) {
+            tok = tok->astParent();
+            const Token *condTok = tok->astOperand2();
+            // check if condition is possibly false, since template arguments are not set to known
+            if (isPossibleBool(condTok, false)) {
+                tok = tok->link()->next();
+                if (Token::simpleMatch(tok, "{"))
+                    tok = tok->link();
+                else while (tok && !Token::simpleMatch(tok, ";"))
+                        tok = tok->next();
+                continue;
+            }
+        }
         if (Token::simpleMatch(tok, "try {"))
             tok = tok->linkAt(1);  // skip till start of catch clauses
         if (tok->str() == "throw")
diff --git a/test/testexceptionsafety.cpp b/test/testexceptionsafety.cpp
index 0c43cb6c689..2772fbb80dc 100644
--- a/test/testexceptionsafety.cpp
+++ b/test/testexceptionsafety.cpp
@@ -361,6 +361,27 @@ class TestExceptionSafety : public TestFixture {
         // avoid false positives
         check("const char *func() throw() { return 0; }");
         ASSERT_EQUALS("", errout_str());
+
+
+        // #11691: FP throwInNoexceptFunction with if constexpr in template
+        check("template<bool IsNoThrow>\n"
+              "static void* MyNew(size_t Size) noexcept(IsNoThrow) {\n"
+              "    void* Memory = std::malloc(Size);\n"
+              "    if (!Memory) {\n"
+              "        if constexpr (!IsNoThrow) {\n"
+              "            throw std::bad_alloc();\n"
+              "        }\n"
+              "    }\n"
+              "    return Memory;\n"
+              "}\n"
+              "void* AllocateMemory(size_t Size, bool UseNoThrow) {\n"
+              "    if (UseNoThrow) {\n"
+              "        return MyNew<true>(Size);\n"
+              "    } else {\n"
+              "        return MyNew<false>(Size);\n"
+              "    }\n"
+              "}\n");
+        ASSERT_EQUALS("", errout_str());
     }
 
     void unhandledExceptionSpecification1() { // #4800