The Italian giants have been following a deal for the central midfielder since January and have finally confirmed the signing.
He has signed a four -year -old deal with AC Milan who would keep him in the club until the summer of 2029, where the club has the option to extend another year.
Ricci, 23, is highly regarded in the Serie A and is seen as a ready-made replacement for Tijjani Reijnders, who came to Man City earlier in the summer.
Luka Modric will also become a member of the club after the FIFA Club World Cup campaign from Real Madrid, while last season they want to improve their eighth-placed competition finish.