The German side are hoping to reach an agreement that would see Cancelo move immediately on an initial loan with an option for a permanent deal in the summer for €70 million.
Sources have told ESPN that the player's relationship with manager Pep Guardiola has broken down, and Cancelo is keen to leave after growing frustrated with his lack of playing time at City.
The 28-year-old was a regular in Guardiola's team at the beginning of the season but has started just five of the past 14 games in all competitions.
A dip in form also saw him lose his place for Portugal during the World Cup in Qatar.
City have not offered Cancelo to other clubs, but Bayern's bid is thought to represent good business for the club and the player.
Cancelo arrived at the Etihad Stadium from Juventus in 2019 and was key to Premier League title wins in 2021 and 2022. Last season, he made 52 appearances in all competitions and featured in 36 of City's 38 league matches.
He was also named in the Premier League team of the year in both 2021 and 2022.
Sources told ESPN that City are unlikely to try to replace Cancelo if he moves to Bayern before the deadline with Guardiola feeling he has enough cover at full-back and centre-back to cope without him during the second half of the season. City remain in the hunt for trophies in the Premier League, Champions League and FA Cup.