Reports emerged earlier today that Moyes was on the verge of being sacked by the Red Devils after a hugely disappointing first season in charge and suggestions that the players had lost confidence in him.
Neville admitted that he fears for the former Everton boss but believes that United should stick with him rather than succumbing to the increasing culture of sacking managers.
"I fear for him. I haven't spoken to anyone at the club today. Usually they don't comment on anything, but they're going to have to clarify this position quickly. They might have to break rank and do what they don't normally do and make a statement pretty quickly," Neville told Sky Sports News.
"I genuinely believe when you give a man a six-year contract, that he deserves the opportunity and time, and professionally the respect. Football is a world of madness. The average manager gets sacked every 12 months and I've always felt Manchester United should be different, hold themselves up as a club that basically stands against what's happening in the game.
"We used to laugh at Italy 20 years ago and say it's ridiculous what's happening there. Now we've almost become accustomed to it. I'm hoping the club will come out and clarify it, and maybe it won't be true."
United are currently seventh in the Premier League table, 13 points off the top four and 23 behind leaders Liverpool.