City host Borussia Dortmund on Wednesday knowing failure to beat the Germans will severely dent their hopes of qualifying from Group D.
Mills, who was at City for two years, told the Sunday Mirror: "If Sergio Aguero hadn't scored three minutes into injury-time (against QPR), the chances are Mancini would have been out of a job. That's how it goes. He may not survive if they don't win the Champions League. There's pressure on him to win a trophy.
"He has to at least get to the semi-finals and show progression in the Champions League - and maybe finish in the top two in the league.
"With success comes expectation and pressure. That's what you have to deal with."