Scans confirmed the severity of the injury and the Chile international will fly to Barcelona on Friday (AEST) to undergo further tests.
City revealed the news after Bravo had earlier tweeted: "No matter what happens or how it looks today, life goes on, and tomorrow will certainly be better."
Bravo's unavailability leaves City without an experienced back-up to first-choice goalkeeper Ederson, with Joe Hart sold to Burnley before the transfer window closed.
Indeed, youngster Daniel Grimshaw is the only other goalkeeper in City's first-team squad.
Bravo arrived at the Etihad Stadium from Barcelona in August 2016 for a reported fee of £17million with a view to replacing Hart as the club's number one.
However, after a number of high-profile errors, manager Pep Guardiola sought to find an alternative, with Ederson signed from Benfica ahead of the 2017-2018 title-winning season.
City lost Kevin De Bruyne to a knee injury for three months last week.