There are several entries on this link below. The only one I read requires Leather repair compound, Applicator, Leather color coating, among other materials that you probably have on hand.

I looked on eBay, doing a search for 'leather repair kit' and there are ones for leather and vinyl under $10, with free shipping.

Here is a link that might be useful: How to Fix a Peeling, Bonded Leather Sofa

I have had faux leather shoes that peel. I took it to the shoe repair and he said there is nothing you can do with something like that BUT to help a little, he would glue around the edge where it was loose and polish the whole thing (my shoes were black). Didn't look the best but it was on the side and I am still wearing the shoes now and then. Not sure about your chairs....could an upholster look at the arm of your chair and maybe suggest a new piece of leather.

Very cool - Thank you!

While the repair kit has been used forever, it takes some ability for detail work and color mixing. There are a variety of repair kits available, but I think they will look repaired and can make it look worse than the before.

An arm cover close to the same material may the easiest way to deal with it. This doesn't help though if other areas start pealing.

Perhaps check out a good auto detailer and see what they use or how it is done. Called "Pleather", it is a standard in the auto industry. The shoe repair guy definitely has the answer. Tough to find one here. Most likely uses the same process as the car guys.