Although Van Gaal has guided United to within touching distance of the Premier League summit, he has come in for criticism for the team's style of play this season.
Neville's former team-mate Paul Scholes branded United "boring" while there have been some dissenting voices in the crowd at Old Trafford too.
Annoyed at Van Gaal's possession-based style of play, some supporters have started yelling "attack, attack, attack" at the team and they have also booed them off on a few occasions.
But, crucially for Van Gaal, he is understood to retain the full backing of the United board, with those above him expecting him to see out the remaining 18 months of his contract.
The boardroom belief is that there is no widespread dissatisfaction with Van Gaal among United fans.
Senior figures at Old Trafford have dismissed reports there is disharmony between Van Gaal and his players and they also acknowledge the Dutchman is getting results.
Going into this weekend's round of fixtures, United are one point behind leaders Manchester City.
At this stage last season, Van Gaal's team had lost three times and were 11 points behind table toppers Chelsea.
Naturally, there is disappointment among senior figures about United's precarious position in the Champions League.
They believed United were handed the easiest group and should have qualified top with a match to spare, but the only way Van Gaal's men can guarantee their progression to the last 16 is if they beat Wolfsburg away in their final match on Tuesday.
But, overall Van Gaal has the full backing of the board.
Nevertheless, United also concede they must start making plans for when Van Gaal's contract expires in the summer of 2017.
The United hierarchy is understood to be watching with interest to see how Neville performs in his first management role at Valencia.
United executive vice-chairman Ed Woodward sent the former United captain a good luck message after his shock appointment on Wednesday and the 40-year-old will come into consideration for the job at Old Trafford if he performs well in Spain.
Ryan Giggs' stock has risen because of his development as a coach since being appointed Van Gaal's assistant and the former winger "triple-ticks" some of the requirements the board will set for the next United boss.
Senior sources at Old Trafford also concede they will also discuss more experienced managers when considering who will replace Van Gaal in 18 months' time.
Press Association Sport understands the United board will be happy to sanction more spending by Van Gaal in the January transfer window, although as the Dutchman has conceded himself, it will be hard to land all the club's top targets.
United are seriously considering signing a left-back in January because of injuries to Luke Shaw and Marcos Rojo, but Ashley Cole, who is out of favour at Roma, is not being considered, sources said.
United are willing to pay big to bring in a marquee signing next summer - Cristiano Ronaldo, Gareth Bale and Neymar have all been mentioned as targets - but there is a also a recognition that signing such a big name will be difficult.
One person who has no future at Old Trafford is Victor Valdes
The former Barcelona goalkeeper, who fell out with Van Gaal after he accused him of not wanting to play for the Under-21s last season, is free to leave in January.
Source : PA
Source: PA