It can be, but beware of cross-contamination
If you're just starting off on a gluten-free diet, it can be hard to know which grains are safe to eat and which might cause discomfort. Wheat is definitely bad, while quinoa and rice are fine—but is oatmeal gluten-free? The short answer is yes. Technically, oatmeal is gluten-free. According to experts at the Celiac Disease Center at the University of Chicago, "Oats are not related to gluten-containing grains such as wheat, barley and rye," and so oats do not contain gluten. And there are plenty of recent scientific studies that have discounted the role of oats in causing celiac disease. But that doesn't mean folks with gluten intolerance should go ahead and indiscriminately eat all of the oatmeal.
The folks at Quaker Oats explain that even though oats are naturally gluten-free, they "may come in contact with gluten-containing grains at the field, during storage or during transportation. And, as a representative from Bob's Red Mill explained to The Kitchn, oats might also come into contact with gluten during processing, since they're often cleaned on the same equipment as wheat: "Wheat berries and oat groats are incredibly similar, making it challenging for basic grain cleaning equipment to remove all wheat from a crop of oats." This kind of cross-contamination between grains can cause flareups among people who suffer from a gluten intolerance, even if the oats themselves are ostensibly gluten-free.
The general consensus is that if you do have a sensitivity or intolerance to gluten, it's better to purchase packages of oatmeal that have a gluten-free label. Gluten-free oats aren't a different type of oat. They're just oats that have been processed on gluten-free equipment and abide to the US government's definition of "gluten-free." For Quaker Oats, that's done by performing, "multiple quality checks throughout the milling process up through finished product testing so that our millers can confidently produce a gluten-free oat product that meets FDA standards."
If you can't find gluten-free oats and still want to enjoy a hot bowl of oatmeal in the morning, the experts at the Celiac Support Association recommend purchasing steel-cut oats since they are, "processed through a burr grinder which is easier for the processor to clean thus reduce cross contamination." The CSA, which is America's largest non-profit celiac disease support group, still recommend rinsing those steel-cut oats in water before cooking, however, to "remove grain dust that contain residue of other grains."
Ultimately, oats are safe to eat for breakfast—as long as people with a gluten intolerance or sensitivity are following common-sense rules for gluten-free dining.