The 34-year-old goalkeeper was hurt during last week's Manchester derby at Etihad Stadium, which finished 0-0.
Willy Caballero is now expected to play for the rest of the season as City look to hold on to a place in the top four and Champions League qualification.
Boss Pep Guardiola also confirmed striker Sergio Aguero will miss Saturday's game with Crystal Palace.
The Argentina international injured himself against Middlesbrough, but could return for their next Premier League match on 13 May.
"Aguero is out. Hopefully he'll be back for Leicester, but this game no," said Guardiola.
"John Stones and Bravo are out, Claudio until next season and John maybe in the next games [can return]."
Guardiola also confirmed David Silva will miss the Palace game.
Manchester City's Claudio Bravo will miss the last four games of the Premier League season with a calf injury.