Vegetarian gluten free savouries
Pure oats ARE gluten-free. If you can't tolerate gluten at all, you need to be sure that your oats aren't &quot;processed in a facility that also processes wheat, rye or barley&quot;, which can contaminate oats with gluten. The big brands in the US that guarantee gluten-free are Bob's Red Mill and Trader Joe's.
Check the label if you are sensitive to gluten!
Quaker specifically does not make a gluten-free claim for their oats:

The oats we buy are handled and transported in bulk by our suppliers. Cross contact can occur if the oats are grown in fields or transported in vehicles that once contained other grains. Since the kernels of the other grains are similar in size, shape and color to the oat kernel, it is almost impossible to separate them.
Although wheat, rye and barley are not part of the ingredients in Quaker Steel Cut, Quaker Old Fashioned, Quaker Quick Oats, Corn Bran Crunch, Grits and Cornmeal there is the possibility that they could contain trace amounts of these grains.  For these reasons, we've never claimed these products are gluten free.

Quaker Oats
EDIT: I just saw a commercial advertising that Quaker is now making a gluten free claim on their oats (although they have not yet updated their website). So check the label if you need no gluten in your oats.