The 22-year-old will join Antonio Conte’s side for a fee of around €25 million according to Italian journalist Gianluca DiMarzio.
Musah is one of several Milan players expected to leave this summer following their failure to qualify for European football last season.
Napoli have also decided against activated the option to buy included in Philip Billing’s loan deal from Bournemouth.
Billing will now return to the Premier League side with Napoli set to sign Kevin de Bruyne on a free transfer soon.