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.
A legal category rather than an ingredient. Under 21 CFR 101.22 a natural flavor is an extract, essence, distillate or similar derivative of a spice, fruit, vegetable, yeast, herb, root, bark, dairy or meat, whose significant function is flavouring rather than nutrition. One line on a label can stand for a dozen substances, and the rule exists because flavour formulas are trade secrets — the manufacturer is not hiding anything from you specifically, they are hiding it from their competitors.
Because it is the one place on a US ingredient label where an honest, careful reader can still get it wrong. People learn to scan for wheat, barley and malt, find none, and conclude the product is fine. For most ingredients that reasoning holds. Here it does not, and the reason is a gap in allergen law rather than anything the manufacturer did.
A gluten-free claim resolves it completely and nothing else does. The claim covers the finished product including its flavourings, so the manufacturer has taken responsibility for what is in there and is subject to the 20 ppm limit in 21 CFR 101.91. Without that claim, the honest position is that the label cannot tell you — not that the product is unsafe. Most natural flavors are not barley-based. The point is that you cannot establish which ones are by reading more carefully, so for a product you eat often it is worth one email to the manufacturer rather than a recurring guess.
This is why "cannot verify" exists as a verdict. When CiCi sees natural flavors with no gluten-free claim on the package it does not guess, and it does not quietly treat the absence of the word barley as evidence — it says the label does not resolve this and explains why. When a gluten-free claim is present it clears, and says which fact cleared it.
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.