It depends
This is the most important thing to understand about US labels. Wheat inside "natural flavor" must still be declared, because wheat is a major food allergen. Barley and rye inside "natural flavor" need not be declared at all — a natural flavor can be built on barley-grown yeast extract and the label will never say so. Without a gluten-free claim on this package, there is no way to resolve this from the label.
Because the answer depends on the rest of the label, scanning the actual product is the only way to resolve it — CiCi checks for a gluten-free claim, the allergen statement, and which agency regulates that label, then tells you which way it lands.
flavors is covered by CiCi's entry on natural flavors, which share this answer and the reasoning behind it — the full explanation is there.
Every one of these is the same ingredient. The number of names is a large part of why gluten slips past people who read every label.
Point CiCi at the barcode on the product you're holding and get this same reasoning in about two seconds, plus the carbs and the full nutrition panel. 360,000+ packaged foods are already read and waiting.
This answer comes from CiCi's ingredient lexicon — the same rules the scanner uses, not a separate article. Where reputable sources disagree, CiCi takes the conservative position and says so.