Sarri has been under constant pressure in the past few months with Chelsea failing to impress consistently.
The Blues needed two late goals to turn the match around against relegation-threatened Cardiff last week with Chelsea fans voicing their disproval at Sarri.
Chelsea backed that turnaround up with a comfortable victory over Brighton in midweek but speculation surrounding Sarri’s future remains.
And Neville believes the way the fans are reacting could be a sign of things to come for Sarri.
“It was an awkward situation to hear the fans turn on him like they did,” former Manchester United star Neville said during Sky Sports’ Monday Night Football.
“I think it was dwindling away that he was potentially going to get sacked then all of a sudden the players come back and turn it on its head.
“He has to come out after the game, do the interviews and try to make sense of it all.
“I’ve been there – in purely football terms there’s nothing worse than when fans turn on your own players and manager. I’ve been there and witnessed it.
“I’m not quite sure how he must be feeling in the sense that he must know he’s so finely tuned at Chelsea that one bad game, one bad substitution that the fans don’t like and the hardcore, the away fans, have turned on you as well, it’s not just the home fans.
“I just feel it’s going to resurface at some point and I’m quite sure what’s going to save him at the end of the season.”
Neville’s Sky Sports colleague Jeff Stelling believes Sarri needs to be more flexible if he is to succeed.
He told Express Sport: “At Chelsea they’ve got so many options, not that Sarri ever wants to use them. Sarri is stubborn with a capital S.
“He seems to be totally set in his ways and will not bend, he’s not remotely malleable.
“He pays no heed to pressure from the press, fans or anyone else for that matter.
“In some respect it’s admirable, but I think you’ve got to make sure, when that is your approach, that you’re getting it right in the first place.
“If you’re not getting it right and then refuse to change, it’s not quite so admirable.”