City have struggled to match the success of winning the Premier League last season as they trail rivals Manchester United by 12 points in the title race.
"Mancini is under pressure," Mills told BBC Sport. "It looks like they have blown their chances of retaining the title.
"You wonder what would happen if they went out of the FA Cup against Leeds. I think Roberto Mancini goes."
City's last hope of winning silverware this season hinges on the FA Cup when they host Neil Warnock's side in the fifth round of the competition on Sunday.