Shortly after the announcement, Pogba posted to his social media channels a photo of himself making a mischievous smirk, alongside a comment reading "caption this".
Former United captain Neville said on Sky Sports: "When I look at the values of Manchester United, you look at Mauricio Pochettino's belief in young players at (previous club) Southampton and with Tottenham".
However, given the tense relationship between Pogba and Mourinho, Neville felt the midfielder was guilty of a callous piece of opportunism.
Manchester United's share price increased by almost six per cent after Jose Mourinho was sacked as the club's manager.
Sunday's loss to the Reds was seemingly the final in the coffin for Mourinho, who leaves after two-and-a-half-years in charge at Old Trafford.
"I do believe that dancing on the grave of the manager who has just been sacked is out of order".
"There is probably a high level of dislike there, but Jose Mourinho lost his job this morning and Paul Pogba should not have allowed that to happen".
Official: Jose Mourinho sacked by Manchester United
"I think he would have liked to have gone public earlier on Jose, and how the club is at this moment in time".
"Jose Mourinho doesn't think Paul Pogba is as good a player as he thinks he is".
Ed Woodward made a decision to sack Jose Mourinho after the humiliating 3-1 defetat to West Ham in September and didn't bother making contact with any of the Portuguese's January transfer targets.
"As manager, you know full well you're the one who carries the can", Neville said.
But experts say it is the players who are the real stars, not the manager, and that when a team like United struggle as they have this year the pressure to make changes will come from the fans, shareholders and even the players.
As a manager, he had a disastrous eight-month spell in charge at Cardiff City in 2014 before returning to Molde in 2015 and has just signed a new three-year deal until the end of the 2021 season. "In my mind as well, I never thought for one second Jose would leave without winning a title".