The 17-year-old goalkeeper is out of contract and free to leave the Parkhead club at the end of the season.
He has yet to be involved with Celtic's first-team but has attracted plenty suitors with his performances in the youth ranks, particularly at the Marc Overmars Tournament in 2017 where he was voted Best Goalkeeper in the competition.
According to reports in Italy, Serie A giants Milan view Angelini as a potential successor to Gianluigi Donnarumma.
Italian rivals Lazio and Atalanta are also believed to be keen, with Man United, Everton and Southampton also reported to be competing for his signature.
It would be a blow for Celtic should another up and coming prospect leave the club having lost Liam Morrison and Barry Hepburn to Bayern Munich, Liam Hughes to Liverpool and Josh Adam to Manchester City in recent years.