The oats do not contain gluten. However because these are a major agricultural product they are handled by the same mills and processing plants, they might get a little quantity. However in general, they are gluten free. Don't worry.
Many oats are grown in the same fields as wheat and often contain gluten, Bob's Red Mill has gluten-free oats as does some other brands...I would check them out. Don't take a chance on regular oats...get gf.
To be totally safe, you need to eat organic gluten-free oats. Oatmeal purchased at the store has generally been processed in the same facility that processes wheat and there is cross-contamination.
Do corn & oat contain gluten.