The former Manchester United defender has been critical of Pellegrini's side in recent weeks as the reigning Premier League champions have seen their chances of retaining the crown slip following a dismal run of form and results.
But, despite the Chilean being the favourite with many bookmakers to be the next managerial casualty and Jurgen Klopp available in the summer after announcing he is to leave Borussia Dortmund, it is understood City chiefs are not entertaining the idea of replacing Pellegrini at present.
And Neville, talking at a Budweiser Dream Goal event in Regent's Park, believes keeping faith in Pellegrini would bring some much-needed stability to the Etihad Stadium.
"I hope they don't change manager," he told Press Association Sport.
"I hope that Manuel Pellegrini stays at Man City and I hope he is able to correct the things that have gone wrong in the last four months.
"He used the term 'garbage' in recent weeks when discussing City's form and four wins in 15 games for a club like Man City is not right and I hope he stays to correct the performances and can complete the job."
A 4-2 derby defeat to United last weekend leaves City fourth in the table, with qualification for next season's Champions League not yet assured.
Just six wins since the turn of the year has seen the club drop out of the FA Cup to Middlesbrough and come up short against Barcelona in Europe for the second season in succession.
Neville feels some work on the side in the summer is required but reckons the type of wholesale changes being suggested may not be the way to go.
"I don't rule anything out - in football you don't make wild predictions," he said when asked if the current City team will struggle to get back to their best.
"These players aren't finished, this team aren't finished
There needs to be some renovation to their house - the people there know that, they don't need me to tell them.
"I have spoken a lot about City over the past months
I have praised them over the last three or four years and rightly so
Some of the football I have seen them play in that time has been the best and their players have been outstanding.
"They have come to a time now where they have dropped below the standard they have set
They have set a standard I have watched, commentated on and been asked to speak about
I have enjoyed when I have watched them
But now, I'm not enjoying watching them."
Dietmar Hamann, meanwhile, believes Klopp would be the ideal candidate to succeed Pellegrini should City's owners opt for a change in the dugout.
"What Klopp has done at Dortmund is second to none
If there is a shortlist at City, he should be up there," Hamann, who played for City between 2006 and 2009, told talkSPORT.
"He is very well respected, he is articulate, and he speaks very good English
He doesn't hold back - he says what he thinks
There are no guarantees with anyone but he deserves a shot at a top club in any country.
"He has got the fire, because he is a bit younger than Pellegrini, to make him aware of what is at stake every weekend.
"You have got to have a relationship with the players these days or you won't succeed and he will be very good at that
He will be very good at getting players and people on board and that is why I think he would be a success.
"As I said, there are no guarantees anywhere, but I think he has got more right to take over Manchester City than Pellegrini ever had."
:: Budweiser Dream Goal, a hunt to find the nation's best amateur goal scorers - www.budweiser.co.uk/DreamGoal
Source : PA
Source: PA