City trail Premier League leaders Manchester United by 12 points and were knocked out of Europe before Christmas.
"Mancini is under pressure," said Mills. "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."
"Manchester United's 12-point lead is too much," Mills told BBC Sport. "I think Mancini will go in the summer anyway if they don't win the league.
"But, if they go out of the FA Cup at this stage, I could see the owners making a decision sooner than that."