The 26-year-old Italian goalkeeper is a product of Inter’s youth system but he never made his debut for the club, bouncing around lower division sides like Avellino and Pordenone on loan before making a €4.15m move to Monza last year.
Di Gregorio has impressed since making the move to Monza, keeping 43 clean sheets across 124 matches across all competitions, regularly producing important saves.
As reported by Tuttosport via TMW, Milan and Napoli have both been keeping tabs on Di Gregorio, with the former identifying him as a possible option should a mega-offer arrive for Mike Maignan and the latter believing he could be a replacement for Alex Meret in the future.
Roma were interested in the Italian shot stopper in the summer but a €20m price tag pushed the club away.
Unnamed clubs from the Premier League are also interested in Di Gregorio and could try to make a move next summer.