Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
Yet the Italian is under pressure following the Blues' European exit, their defeat by Borussia Dortmund on Tuesday meaning they have not even landed in the Europa League following their Champions League exit, as was the case 12 months ago.
Neville believes an FA Cup and Premier League underline the progress City have made over the past two seasons.
However, with a new chief executive to impress in Ferran Soriano and a new director of football to work with now Brian Marwood has made way for Txiki Begiristain, he feels Mancini's position cannot be described as secure, especially with Pep Guardiola and Jose Mourinho likely to be looking for work in the summer.
"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," said Neville.
"Whilst Roberto Mancini continues to be successful he should stay in his job but it does feel like one day Mourinho or 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."
Neville was speaking alongside journalists at a Football Writers' Association Forum in Manchester ahead of Sunday's derby clash between City and United at the Etihad Stadium.
The presence of two such prominent former Barcelona figures at the Etihad Stadium has lent credence to the theory Guardiola will choose City as the place to end his year-long sabbatical.
Mourinho has long been touted as Sir Alex Ferguson's successor at United.
And with reports emerging from Spain the former Chelsea boss will leave Real Madrid in the summer, it has been suggested Ferguson might be persuaded to stand aside after spending 26 years at the Old Trafford helm.
Neville simply does not buy into that theory.
"Whilst he wants to carry on, Sir Alex Ferguson will not stand aside for anybody," said the former United full-back and now respected TV pundit.
"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."
With both clubs moving clear of third-placed Chelsea, United and City appear to have the Premier League title between them once more.
Sunday's match is delicately poised, with the Red Devils three points ahead of their neighbours, who have a better goal difference, meaning victory on Sunday will take them top, exactly the same position that existed ahead of the corresponding fixture last term.
"Sir Alex Ferguson has been openly concerned about United's defending," said Neville.
"It must be massively frustrating for him and he would have been embarrassed by the first half against Reading.
"But, at the really big points in the season, do his players let him down? More often than not, they do not.
"But there is also a lot I like about Manchester City. They have dug out results, never seem to go away and are still only three points behind."