The future of the United manager was thrown into doubt on Friday when a report from the Mirror suggested that he'd be sacked irrespective of whether the club beat Newcastle United at Old Trafford on Saturday.
Neville, who played his entire club career at Old Trafford, questioned the "values and principles" of the club if Mourinho were to be sacked.
"The people in the boardroom at this moment in time who are, if you like, overseeing the football operation, are nowhere near good enough".
United's attack did the defence no favours in the 50th minute, with Pogba denied by Newcastle Martin Dubravka from 25 yards despite a deflection and Matic missing the second chance, on an open net, by sending his attempt over the crossbar.
Much has been made of the fractured relationship between Mourinho and star midfielder Paul Pogba, and the World Cup victor has hardly covered himself in glory in recent weeks with his uninspiring performances for the side.
Some reports suggest that United officials could show Mourinho the door during next week's worldwide break.
Another with Premier League experience at one of England's biggest clubs, Antonio Conte has the distinct advantage of already winning the league with the majority of a squad that Mourinho brought to Chelsea.
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Press Association Sport has been told that patience is wearing thin within the club's hierarchy but Friday evening's story appeared to catch United on the hop when contacted.
Jose Mourinho has singled out Marcus Rashford and Scott McTominay for their failure to deal with the current pressure at Manchester United.
Asked who was to blame for United's predicament, Mourinho responded: "The same people that were to blame for the fantastic season of last season".