I suspect the floor is sheet vinyl and not actual linoleum (which is made from cork and varnish pressed together onto a backing cloth). That detail probably isn't very important here.
My suspicion is that the actual problem is the underlying subflooring breaking down from being wet, in which case the only successful repair would be to pull it up and replace it. Particle board and similar products often don't need to be wet very long before they start to break down and the wood bits expand. I could, of course, be all wrong in my guess, though.
If it's clear that there's a void between the vinyl and the subfloor, I'd probably try poking a tiny hole with a pin or needle and applying pressure and moderate heat. I don't think a glue you could inject would work very well; the flooring is usually applied with a sort of mastic compound, relatively thick. Possibly Gorilla Glue would be something to try if heat alone fails, since it tends to foam and expand a little bit.