The Ivorian, who joined the Red Devils from Villereal in 2016, has made 113 appearances for the club since his arrival, scoring once.
TUTTOmercatoWEB report Milan are interested in signing the centre-back on loan this month.
No details as to whether this would be a straight loan or loan-to-buy deal have been disclosed.
What is Bailly’s standing at United?
Bailly has had an interesting career in Manchester, suffering several injuries and failing to establish himself as a first-choice name.
The 27-year-old has slipped further down the pecking order at Old Trafford since the arrival of Raphael Varane in the summer and a move to the Italian capital could be just what he needs to get game-time.
He signed a new contract with United in April last year, keeping him at the club until at least 2024, unless sold.
Bailly is currently away on international duty competing in his third Africa Cup of Nations (AFCON) tournament with Ivory Coast.
His absence could mean a potential departure from the Red Devils during the January window will have to wait until his return from the tournament.
Ivory Coast’s final group stage game is on 20 January and if they fail to reach the next stage of the competition, Milan would then have the opportunity to pursue the centre-back’s signature.
However, there is potential for the nation to go all the way to the AFCON final, which would mean Bailly remains on international duty until at least 6 February, thus making any potential move away from United this month a hard task.