Bravo was one of Manchester City’s first signings for Pep Guardiola and the Chilean quickly replaced Joe Hart as the first-choice goalkeeper.
Brazilian international Ederson has emerged as the new number one, though Bravo managed 17 appearances last season.
The 37-year-old is wanted by new Real Betis manager Manuel Pellegrini, with Goal.com reporting that the ‘keeper is willing to take a paycut to facilitate a move.
Bravo is best known for his time in Spain as Barcelona’s goalkeeper, when he shared duties with Marc-Andre ter Stegen before being forced into the reserve role.
The Chilean was also linked with moves to Arsenal, Celtic, Galatasaray and City-affiliated group New York City FC.
Bravo will remain with the club until the conclusion of their Champions League involvement, with the side due to play Lyon in Lisbon as part of UEFA’s mini end-of-season tournament.