It's not clear to me what "entire area" area you want leveled.

Also why do you want it leveled?

You can see most clearly in the back that there is a depression where the edging was. I'll be removing the edging from the front as well. The entire area within the circle is a little lower than the grass outside, but mostly I just want to even it out so it's not a tripping hazard.

Maybe I don't quite understand what you want to do, maybe level the area in the circle? I think you can do that unless you find you are harming shallow roots as you work with a hoe or whatever. One thing I would avoid, however, is changing the soil level around the bottom of the trees, where the tree goes into the ground. I'm no expert, but that's my 2 cents.

Hard to tell how much leveling would need to be done to get it where you want but a few inches of soil/compost should be fine. If it was my project I would do compost and mulch. Grass should be okay but the trees would prefer mulch as they would get more water and air to their roots. Cottonwoods are notorious for wanting water.

Thank you all for your perspectives. Sounds like it won't be too much of an issue if I add a little soil, but would be better for the tree to stick to mulch and that should help fill the area too.

Gg's answer is correct.

FWIW a moronic neighbor had his "best landscaper in the county"* pile what looked like 4 to 6 inches of top soil on top of most of the root zone of a junk tree, probably a norway maple. To grade it to match some hardscaping he had done. Within a couple years, parts of the canopy were dying. But the hideous thing struggled on - it was almost a decade before he got tired of having said "best landscaper" trim dead branches out of the canopy. Point is, changing soil level on top of root zone is a bad idea, but it can cause only a slow death in some cases. (and probably a fast one in others)

* PS - a laughable accolade - I had one encounter with this bonehead and decided never to utilize his services. He had a bad attitude and seemed likely to be aggressively ignorant about some topics - which he obviously did prove himself to be, some years later, about grading over the roots of a mature tree in the neighbor's yard. I'll always wonder if that was the same guy who cut down a monkey puzzle at an Edwardian-era estate in New Castle County, Delaware, one of the only on the east coast, for "not looking English enough"! Perhaps the most risible 'anecdotes indicting an entire profession in a part of the world' I know of. We lead the world in many, many areas, but overall horticultural savvy is not one of them. As it would take any observant visitor to the suburbs of a major city in Australia, New Zealand, the UK, FR, or DE only a few minutes to determine. (Slivers of certain areas of the US do a little better, like the PNW)

Form what has been said and looking at the picture it looks like your will need to add a little soil in a couple of low spots and then level them off. From what I have heard It does not seem that your "leveling" will raise the general level of the area around the trees.

It what I understand is your plan, go ahead an do it. It will not hurt the trees. From the picture it looks like the area gets plenty of sun so you may try to plant grass and see what happens. However unless you have a small yard and use a push type of mower, you may have trouble getting a riding lawn mower between the trees.

Personally any area of the yard that I have problem growing grass becomes a flower bed. So I probably would mulch the area without edging and plant tulips, glads, and similar plants that would provide color year around.