The Italian's position as coach at the Etihad is under threat thanks to a misfiring title defence, which sees City languishing 12 points behind league leaders Manchester United with 12 games left to play.
Toure has admitted that winning the cup will strengthen Mancini's position at Eastlands and help salvage the club's faltering campaign.
The Ivorian centre-back said: "Winning a trophy will help the manager to be here, for sure.
"For a team like City it's difficult to go a season without winning one trophy. So the FA Cup is really important for us."
City booked a quarter-final tie against Barnsley next month by beating Leeds United 4-0 at the Etihad on Sunday.