The former Arsenal youngster now has less than two years left to run on his deal with the Scudetto winners and has been strongly linked with an exit in recent months.
Bennacer was a driving force in Milan's run to the Serie A title last season, but Stefano Pioli's side could come under pressure to sell if he does not commit himself to a new deal.
Arsenal were briefly credited with an interest in their former academy player over the summer, while Liverpool were also tipped to make a move amid their search for midfield reinforcements.
Now, Calciomercato claims that Man United have made contact with Bennacer's agent Enzo Raiola as Milan try to tie the Algeria international down to a new contract.
Bennacer's existing deal with the Rossoneri reportedly sees him earn just €1.5m (£1.3m) per year, but Milan are willing to increase his wage to as much as €4m (£3.4m).
However, the Italian champions are yet to reach an agreement with Bennacer's entourage, and Man United would likely be able to offer him a more lucrative financial package.
The report adds that Man United are facing competition from Chelsea, who are also attempting to prise Rafael Leao away from San Siro, for the signature of Bennacer in 2023.
Bennacer spent two years in the Arsenal academy ranks between 2015 and 2017, but he only made one senior appearance in an EFL Cup defeat to Sheffield Wednesday.
The Algerian left the Gunners for Empoli five years ago and helped the Azzurri achieve promotion to Serie A in the 2017-18 season, although they were relegated the following year.
Nevertheless, Bennacer's performances caught the eye of Milan, who paid €17.2m (£14.7m) to secure his services in 2019, and he has since recorded four goals and eight assists in 126 appearances for the Rossoneri.
The 44-cap Algeria international has remained a key player for Pioli this term, scoring once and providing one assist in 21 matches and starting all six of their Champions League group-stage games.
Man United lost the likes of Paul Pogba, Nemanja Matic, Juan Mata, James Garner and Andreas Pereira over the summer, but Erik ten Hag responded by bringing in Casemiro and Christian Eriksen.
The Red Devils are expected to be in the market for fresh faces in the middle in 2023, as Donny van de Beek is facing an uncertain future at Old Trafford and could potentially leave to open up a space for a new signing.
Borussia Dortmund's Jude Bellingham, Real Sociedad's Martin Zubimendi and Brighton & Hove Albion's Moises Caicedo are also believed to be on the Red Devils' midfield shortlist for next summer.