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Bravo has dropped into the free agency pool after reaching the end of his contract at the Etihad Stadium.
The 37-year-old joined City from Barcelona for an initial£15million (£20m) fee in the summer of 2016, and went on to appear in 61 games for the club, recording 20 clean sheets in the process.
Bravo won six major trophies during his time in Manchester, including two Premier League titles, but had to be content with a back-up role in the squad following Ederson's arrival in 2017.
City have confirmed the veteran shot-stopper's exit in astatement via their official website, which reads: "Claudio Bravo has left Manchester City after four years with the club.
"Bravo will be best remembered for bringing a new style of goalkeeping to City – the so-called ‘sweeper-keeper’, with his excellent technique and control with the ball at his feet.
"City fans will also recall his penalty shoot-out heroics, first against Wolves at the Etihad in 2017-18 and again when his save from Liverpool’s Georginio Wijnaldum secured the 2019 Community Shield.
"Everybody at Manchester City would like to thank Claudio for his four years at the club and wish him good fortune in his future endeavours."
Goal has reported that Bravo will embark on a new challenge with Betis in 2020-21, with negotiations over his salary still ongoing as he prepares to accept a significant pay cut atBenito Villamarin Stadium.
Los Verdiblancos' newly-appointed head coachManuel Pellegrini has requested for the Chilean shot-stopper to be brought in as he aims to shore up a defence that conceded 61 goals in La Liga last term.