Mourinho was axed on Tuesday following their dismal start to the season that leaves United 19 points off the top of the Premier League and 11 off the pace in the race for the top four.
Pochettino is reported to be one of United’s top targets but they have revealed Michael Carrick will take over for the next two days before being replaced by a temporary manager who will guide them to the end of the season.
And Neville has urged them to do everything they can to get Pochettino when they appoint a permanent successor.
“I’ve been fortunate to spend time at Tottenham’s training ground, and for me he feels like the ideal candidate,” the United legend said.
“Man United have tried managers who have won European cups, they’ve tried managers who have won multiple leagues.
“My view is they need someone who meets the three key principles of the football club, and that is promotion of youth, entertaining football, and to win football matches.”
Despite Mourinho leaving United in a sorry state after just over two and a half years at Old Trafford, the club thanked him for his service in a club statement after breaking the news of his sacking.
The Portuguese won the Europa League and League Cup in his first season and led them to a second-place finish in the Premier League last term.
“Manchester United announces that manager Jose Mourinho has left the club with immediate effect,” the statement read.
“The club would like to thank Jose for his work during his time at Manchester United and to wish him success in the future.
“A new caretaker manager will be appointed until the end of the current season, while the club conducts a thorough recruitment process for a new, full-time manager.”
Pochettino is in the second favourite to take over at the end of the season according to bookmakers.
However, they say former Real Madrid boss Zinedine Zidane is the frontrunner as it stands.