After an error-strewn start to the season, the England international has lost his first-team slot at the Etihad Stadium.
Even though he is thought of as one of the Premier League’s top stoppers, Pellegrini is believed to be unconvinced by Hart, and is ready to bring in a new ‘keeper over the winter.
Bravo has emerged as one of the Chilean’s top targets, with the ex-Malaga boss a big fan of his compatriot.
The South American was a key member of Real Sociedad’s Champions League qualification winning campaign last season, and has remained an ever-present this term in Europe’s top club competition.
Barcelona have been alerted by the 30-year-old’s displays, but Pellegrini is hopeful that he can lure him to Manchester.
Bravo is thought to satisfy City’s desire for a commanding and reliable presence between the posts, with the glovesman captain of his country and second in command at club level.
The player himself has a £25m release clause in his current contract, but it is thought that he may be allowed to move for a much lower fee.