The Ivory Coast international's current deal at San Siro is due to expire in June 2022, meaning that, as it stands, he would be free to enter into pre-contract discussions with foreign clubs at the start of next year.
Man United boss Ole Gunnar Solskjaer is seemingly keen to bring a new central midfielder to the club at some stage in 2022.
However, the 20-time English champions will likely have to hold off competition from the likes of Paris Saint-Germain, Chelsea, Liverpool and Tottenham Hotspur to secure Kessie's signature.
Here, Sports Mole looks at how Solskjaer's side could line up with Kessie if they manage to bring him to the club.
Kessie enjoyed an excellent 2020-21 campaign for his Italian club, scoring 14 times and registering six assists in 50 appearances, and he made a strong impression against Man United in the last-16 stage of the Europa League last term.
The 24-year-old is capable of operating as an attacking midfielder and largely featured in that area of the field for his country at the Tokyo Olympics, but Kessie primarily operates as a defensive midfielder at club level.
Classy on the ball and a powerful runner, Kessie would be a huge asset to Man United, who could potentially attempt to agree a fee with Milan in January in order to sign him for the second half of the 2021-22 season.
Solskjaer might feel more confident playing Paul Pogba as a central midfielder if Kessie arrives as protection, which would allow Jadon Sancho to operate off the left, Mason Greenwood off the right, with Bruno Fernandes continuing behind Cristiano Ronaldo.
Fred and Scott McTominay are favourites of Solskjaer in the middle of the park, but Kessie would be an upgrade on both players, and the Ivorian would offer more protection to a back four of Aaron Wan-Bissaka, Harry Maguire, Raphael Varane and Luke Shaw.