Current Red Devils boss Louis van Gaal's contract expires in 2017 and the club will consider Neville and assistant manager Ryan Giggs for the vacant managerial positon.
The legendary duo won 21 Premier League titles between them during their time at the club with Giggs, who retired from professional football in 2014, briefly taking charge at Old Trafford after David Moyes' departure.
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Therefore, the Welshman is widely considered favourite to succeed Van Gaal in 2017.
Neville, meanwhile, has been given the opportunity of his career after being named the new Valencia boss. He will link up with brother Phil, who is assistant at the Spanish club.
The job is Neville's first taste of football management, though he has been England boss Roy Hodgson's assistant since 2012.
The 40-year-old former right-back left his post as a pundit and commentator for Sky Sports last week as a result of his move.
He watched his new side draw 1-1 with La Liga champions Barcelona on Saturday night and will take charge of his side for the first time on Wednesday, when Los Murchielagos play Lyon in their final Champions League group game.
Depending on how effective Neville is during his reign, United may consider him as a possible long term replacement for Van Gaal in 2017.
Bury-born Neville, who represented United for 19 years, has the backing of former teammates and managers. Paul Scholes and United great Sir Alex Ferguson have both tipped him for success.