The Citizens, who are currently second, have been linked with a move for former Real Madrid boss Manuel Pellegrini at the end of the season.
City are 12 points behind league leaders Manchester United in the title race, while Mancini’s men face Barnsley in the FA Cup quarter-finals.
But Clichy, who was signed by Mancini in 2011, says the 48-year-old deserves credit for ending the club’s silverware drought amid speculation about the manager’s future.
“Only time will tell whether the club decides to sack him or keep him, but he’s doing well,” Clichy told the Sun.
“It’s the people above who decide – but I believe he can take us higher.
“He’s the one who brought me here. Of course I’d like to see him stay. As a player you want to give your best for a manager like him.
“If you look at his record, he’s the one who brought silverware to this club, the FA Cup, Community Shield, Premier League. You have to give credit where it’s due.
“I think people say with that money any manager could do the job but I doubt it.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news