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.
Do Oats Contain Gluten?
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.