Manchester United have not responded to questions about the report in Saturday’s Daily Mirror.
The Red Devils’ sackings of Louis van Gaal and David Moyes – Mourinho’s two immediate predecessors – were both reported in the media before the club’s official announcements.
“There is something rotten to the core of the decisions that have been made over the last four or five years,” said Neville, who added that he did not know if the report about Mourinho was true.
“If I’m the person who employs the person underneath me and he fails, eventually I’ve got to take the responsibility. Eventually I’ve got to stand up and say: ‘Look, I’m not good enough to do this.'”
Neville, 43, made 602 appearances for United, winning 16 trophies.
He suggested Mourinho should have been allowed to sign a central defender in the summer – whether it be Tottenham’s Toby Alderweireld, Leicester’s Harry Maguire or Bayern Munich’s Jerome Boateng.
The former defender added: “Who is qualified in that football club to be able to tell Jose Mourinho that Alderweireld or Maguire or Boateng are not better than the centre-backs that they’ve got?
“I’m furious. It’s been coming for three or four weeks.
“The dressing room is leading what’s happening. The tail is wagging the dog at the football club. Get some control back, get some leadership.”