The 26-year-old has spent the last three years at San Siro and has established himself as a first-team regular for the Rossoneri, making 126 appearances across all competitions.
The former Chelsea defender has been tipped to return to the Premier League amid interest from Manchester United, but Sky Deutschland reporter Florian Plettenberg claims that Bayern have made their first move for the Englishman.
Plettenberg states that Bayern's sporting director Christoph Freund is a "big fan" of Tomori, who has been discussed internally over the last few weeks.
It is thought that a transfer away from San Siro is unlikely for Tomori in January, though, as Milan are not looking to sell one of their star players, but further negotiations cannot be entirely ruled out as Bayern may make a fresh approach for the centre-back later this month.
Tomori, a five-cap England international, is under contract at the Serie A club until the summer of 2027.
Staying with Milan and Stefano Pioli's side are reportedly set to battle with city rivals Inter Milan for the signature of Aston Villa striker Jhon Duran.
The 20-year-old joined Villa in January last year from MLS club Chicago Fire for around £15m, but he has been force to play second fiddle behind star attacker Ollie Watkins.
Duran is yet to start a Premier League game and has instead made 25 substitute outings over the last 12 months, scoring twice in 13 substitute appearances so far this season.
A lack of regular first-team football for Duran has led to speculation about his future, and Football Insider claims that Serie A duo Milan and Inter are both in talks with Villa over signing the Colombia international this month, most likely on loan.
However, a separate report from Sky Sports News claims that Milan must move out a non-EU player before they can consider moving for Duran.
The potential departure of Duran could see Villa step up their pursuit of a new centre-forward as Unai Emery looks to provide greater competition for Watkins.
One player who does look to be heading for the Villa Park exit door is midfielder Finn Azaz, who is reportedly closing in on a permanent move to Middlesbrough.
The 23-year-old was recently recalled by Villa from a loan spell at Plymouth Argyle where he has scored seven goals and registered five assists in 26 Championship appearances this season.
Azaz also spent the 2022-23 campaign on loan at Home Park and was part of the team that won the League One title, scoring eight goals and providing nine assists in 34 league matches.
According to Football Insider, Villa have accepted a £2m bid for Azaz from Middlesbrough and the midfielder is set to travel up to the North East for a medical.
Azaz is said to be close to reaching an agreement over personal terms with Michael Carrick's side, who are also interested in signing Leeds United defender Luke Ayling.