The Italian goalkeeper, aged 26, emerged from Inter Milan's youth system; however, he never managed to make a senior team appearance for the club.
Di Gregorio spent time on loan with lower-division teams such as Avellino and Pordenone before securing a £3.5m transfer to Monza last year.
According to a report from Tuttosport, both Milan and Napoli have been monitoring the goalkeeper in recent months, and it is understood that several unnamed clubs from the Premier League are also interested in Monza's shot-stopper.
The report adds that Milan sees him as a potential alternative for Mike Maignan, while Napoli views him as a prospective replacement for Alex Meret in the coming years.
During the summer, Roma expressed interest in the Italian, but a £17m price tag deterred the club from pursuing the deal.
Di Gregorio has impressed many onlookers since joining Monza, recording an impressive tally of 43 clean sheets in 124 matches across all competitions.
The Biancorossi goalkeeper has kept four clean sheets in 13 appearances this season and pulled off an incredible save to deny Juventus forward Dusan Vlahovic from the penalty spot last week.