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The Portugal international has joined the Bundesliga champion initially until the end of the campaign, with Bayern board member Hasan Salihamidzic confirming the switch may be made permanent after that.
Cancelo was a star performer for City last season but has fallen out of favour this campaign, with manager Pep Guardiola preferring the likes of Kyle Walker, Nathan Ake, John Stones and even 18 year-old Rico Lewis for the full-back roles.
He started more Premier League games than any other City outfield player (36) last season as it won the title, having more touches (3,908) and attempting more passes than anyone else in the league (2,951).
Cancelo told Bayern's official website: "FC Bayern is a great club, one of the best in the world, and it's enormous motivation for me to now play alongside these extraordinary players in a team.
"I know that this club, this team lives for titles and wins titles every year. I’m also driven by the hunger for success. I'll give my best for FC Bayern."
Cancelo has two goals and four assists from 26 appearances in all competitions this season, but has been an unused sub in the three games since City's 2-1 derby defeat at Manchester United.
Bayern has moved for the 28 year-old in an attempt to turn around a poor run of form since the Bundesliga's resumption after the mid-season break.
Julian Nagelsmann's side has drawn all three of its games 1-1, against RB Leipzig, Koln and Eintracht Frankfurt respectively, leaving it just one point ahead of FC Union Berlin at the top of the table.
Bayern will be without Noussair Mazraoui for several week, the Moroccan suffering an infection after a bout of COVID-19.
Benjamin Pavard played at right-back against Leipzig and Koln, with Josip Stanisic replacing him for the Eintracht stalemate.
Cancelo could make his Bayern debut in the midweek DFB-Pokal clash with Mainz.