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And he is set to leave at the end of the season with no sign of a new deal being offered by the Premier League champions.
Bravo has made it clear to the club's hierarchy that he wants regular football after spending the last three seasons as the second choice to Brazilian team-mate Ederson.
City's sister club New York are leading the chase to sign him when he becomes a free agent but there is believed to be interest from other Major League Soccer clubs, while Mexican side Monterrey have also been linked with him.
Bravo, who is likely to start City's FA Cup fifth round clash with Sheffield Wednesday on Wednesday, has made 15 appearances this season but missed much of the last campaign with an Achilles tendon injury.
Kosovo international Aro Muric stepped up in his absence and will now hope to replace him as Ederson's number two next season. However, the 21-year-old has struggled on loan at Nottingham Forest, making his last of five appearances for the Championship side in September and keeping just one clean sheet.
City also have United States international Zack Steffen after he made a ?7 million ($9m) move from Columbus Crew in December 2018. The 24-year-old has impressed on loan at Bundesliga club Fortuna Dusseldorf but has missed their last seven matches with a knee injury.
Meanwhile, Pep Guardiola paid tribute to Bravo's professionalism after City won the League Cup for the third successive year.
The keeper has faced plenty of criticism since joining from Barcelona in 2016, particularly after his first season when he struggled before eventually losing his place to reserve keeper Willy Caballero.
"He was involved in a lot of titles that we won, he's one of the most incredible professional players I've ever seen," Guardiola said.
"In the first year with Willy he was incredibly correct and hopefully the other players can learn from him, what he won with the national team and other clubs and accept his position.
"He's an extraordinary person, he will have success because he's such an intelligent person.
"The 'keepers in a final, they have to save it when it's a tight game and he did it - like Ederson did against Benzema, he did it again. He played against Liverpool in the Community Shield and Carabao Cup. Many titles he was involved in, I never had doubts his quality as a keeper."