Mancini’s side failed to win a game in their Champions League and failed to secure third place and a spot in the Europa League.
Although City have won the FA Cup and Premier League in the last two seasons the Italian has come under pressure to the surprise of the England coach and football pundit.
Neville said: "Roberto Mancini deserves more time but it is not a ridiculous suggestion to think City would not consider making a change if he doesn't win the league.
"Whilst Roberto Mancini continues to be successful he should stay in his job but it does feel like one day [Jose] Mourinho or[Pep] Guardiola will end up there.
"They have a new chief executive and a new director of football who presumably have the ear of the owner.
"It might be three or five years but eventually they will want to stamp their mark on the club and Mourinho and Guardiola have to be pretty high on the list."
By contrast Neville believes his former manager Sir Alex is in a position of almost total security as the two Manchester clubs prepare for their league meeting on Sunday.
"Whilst he wants to carry on, Sir Alex Ferguson will not stand aside for anybody," said the former United full-back.
"He will decide when he doesn't want to do it anymore.
"He is not the type of person who would just step aside so he could watch someone else manage his football team.
"It has been Alex Ferguson's last season for 15 years now. Unless it is the best kept secret in the world, nobody knows other than him."