According to TuttoMercatoWeb (h/t the Daily Mirror), the Spain international has had his head turned by the Serie A side, who invested heavily in their squad in the summer. His contract at the Etihad Stadium is poised to expire in the summer of 2019.
It's noted that while City were keen on getting the playmaker's contract extended, talks have been shelved because of the Rossoneri's interest.
The 31-year-old joined City from Valencia in 2010 and is regarded as one of the best players in the club's history. Silva has been vital in the team's ascension to football's elite, offering a key creative outlet in two Premier League title wins, two League Cups and an FA Cup triumph.
He's still crucial to the team under the guidance of manager Pep Guardiola. Squawka Football summed up his influence this season:
While City have more dynamic and flashier footballers on their books, Silva is the man who conducts this attacking orchestra.
His touch is tight, his dribbling is astute and his passing in the final third is incisive. For a manager such as Guardiola, he's the perfect midfielder.
City aren't short in this area of the field, with players like Kevin De Bruyne, Bernardo Silva and Ilkay Gundogan all brilliant in their own right. Yet Silva brings a composure, class and experience to the team that trio doesn't.
As Sam Lee of Goal noted, it seems as though after a year of working together, Guardiola is getting the best from Silva too:
Perhaps the veteran midfielder fancies taking on a new challenge. Milan are one of the most exciting clubs in European football to be at, as they appear intent on reaching the pinnacle of Serie A again soon.
Even so, Silva is adored at Manchester City and seems to be loving his football under Guardiola. There can't be many teams more enjoyable to play in than the one he's at the hub of.