It was gathered Mourinho, 53, will replace Louis van Gaal, despite United winning the FA Cup on Saturday.
Van Gaal, 64, two seasons into a three-year deal, said last week expectations at Old Trafford were “much too high”.
But Neville, 39, said he “disagrees with that”, adding: “At the moment, Jose is the best man for the job.”
The former England international, who played for United from 1995-2005 and was first-team coach under David Moyes in 2013-14, told BBC Radio 5 live’s Sportsweek: “For as long as I’ve known it you’ve got to be the best at Manchester United.
“Mourinho is a proven winner. United fans want a team challenging for the league title, they don’t want a team finishing in seventh or fifth.”
The Portuguese won three Premier League titles in two spells at Chelsea and has also won domestic leagues in his homeland, Italy and Spain.
Mourinho was sacked in December after champions Chelsea made a poor start to their title defence, losing nine of their first 16 matches, while he was involved in a high-profile dispute with club doctor Eva Carneiro, who was dismissed and pursued a constructive dismissal case against the club.
“When Jose finishes at a club there is a pattern where he falls out with players, gets disgruntled and the team dips,” Neville said.
“I can’t see him staying for too long at United – maybe two or three years seems to be his shelf life at a club – but as long as he brings success United fans won’t be unhappy.
“The pressure of football management and demands from owners and supporters and media now means two to three years in a job is enough.”
United last won the Premier League in 2013 – in Sir Alex Ferguson’s final season in charge – and have since finished seventh, fourth and fifth, missing out on next season’s Champions League.
“He is not the type of manager that will come in and say this is a transitional period and finishing fourth in the league will be enough – that won’t be enough for the United fans,” Neville said of Mourinho, who has also managed Porto, Inter Milan and Real Madrid.
“I am a Manchester United fan and next season I want them to be winning the league. That will be the demands put on him and I am sure he will have the same expectation.”
There have been suggestions that Mourinho’s pragmatic style of play and confrontational personality may not be popular with the United hierarchy and supporters – but Neville says it could bring positive results.
“I know you have to manage a certain way when you’re manager of Manchester United, but if you take away that brashness – and you saw that a little bit towards the end at Chelsea – you lose what Jose Mourinho is about,” said Neville, who won six Premier League titles, three FA Cups and the Champions League with the Old Trafford club.
“He likes to create this siege mentality and that’s what Sir Alex [Ferguson] did for 26 years. He told us in every team meeting that the rest of the world doesn’t like you, wants you to fail, and that was the basis of our motivation in a lot of our games.
“Sir Alex did like to create a cause for the fight, with the media and with other managers. If Mourinho does that at United, the fans will take him to their hearts.”
Van Gaal, 64, was appointed on a three-year deal in summer 2014 and arrived at United having guided the Netherlands to the World Cup semi-finals in Brazil.
The Dutchman has spent £250m on new players, including £36m on France forward Anthony Martial, and brought through several youngsters from the club’s academy, such as 18-year-old striker Marcus Rashford, who has been named in England’s provisional 26-man squad for Euro 2016.
“Rashford has been a breath of fresh air and I think Martial is going to be an absolute superstar,” added Neville.
“They are not too far away – maybe two or three players and maybe this team could be challenging for a league title.”