No, it wouldn't be athlete's foot (fungus.) There is a form of eczema called dyshidrosis which produces little bumps under the skin of the fingers. That's not very significant and usually doesn't need to be treated. I can't think of anything these could be that you need to worry about. If they persist for a month or two, show a skin doctor, just for peace of mind.
Take care.
Dr. Rockoff
I get these little circles under the skin on my finger tip on my left hand ring finger. They first appear as reddish brown and look more brownish or orangeish in color as they come to the surface. I also get similar, but larger spots under my fingernails.
When the spots on my fingertips break the surface they peel the skin and then heal. I had my doctor take a look and she said it was psoriasis. It comes and goes but seems to only affect me in the winter.
The less I worry about it the less it seems to erupt.