Hi, I am breast feeding and noticing a white patch under the skin near my nipple. Could that be milk stuck under the skin or something else?

Thrush. Sounds like it could be a patch of yeast which can give your baby thrush. Be sure to call your doc, also look in the baby's mouth and see if there are white plaques on the mouth. It is very common for mom and baby to pass it back and forth.
Inclusion cyst. It is an inclusion cyst most likely. Your physician can have you see a dermatologist or breast specialist.