The midfielder was one of Milan's standout performers as they lost 3-2 to the Reds in their dramatic Champions League opener at Anfield last week.
Kessie joined Milan on loan from Atalanta in the summer of 2017, before making the switch permanent two years later.
The 24-year-old - who is also a regular for the Ivory Coast - has made 183 appearances for the Rossoneri, scoring 32 goals.
While the Reds have come flying out of the traps in 2021/22, sitting level on points with United and Chelsea at the top of the Premier League table after five games, they didn't bring anyone in over the summer to replace the departing Georginio Wijnaldum - who joined PSG after his contract expired.
United, meanwhile, made big moves in defence and attack over the summer, but midfield remains an area of vulnerability for Ole Gunnar Solskjaer's side - despite their strong start to the campaign.
Kessie is out of contract at the end of the season, so he would be free to negotiate a pre-contract agreement away from Serie A in January - assuming he doesn't sign a new deal between now and then.
Is he likely to? Well, according to Milan legend and technical director Paolo Maldini - who was in attendance at Anfield, where he saw his son Daniel in action - the club "have been talking to [Kessie's] agent for a year and a half". That probably constitutes a saga, doesn't it?
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