Express Sport understands Moyes will take over at West Ham after the Hammers opted to sack Slaven Bilic earlier today.
Stuart Pearce is set to work under the Scot as coach, with Moyes taking training tomorrow.
From the moment Bilic was axed there was talk that Moyes would bring in Neville, who he previously managed at Everton and worked with at Manchester United.
But ESPN say Neville has snubbed the offer of working at West Ham – because he has his sights set on the Sunderland job.
The Black Cats are on the lookout for a replacement for Simon Grayson, who was axed last week.
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Sunderland are currently stuck in a rut with the club rooted to the bottom of the Championship table.
And Neville supposedly wants to be No 1 at the Stadium of Light.
ESPN say that, so keen is he to be a manager, he has chosen not to follow Moyes to the London Stadium.
Meanwhile, Sky Sports’ Tony Gale appeared to confirm Moyes’ appointment earlier today.
“He’s going to know every one of these West Ham players,” Gale said on Sky Sports of the Scot.
“This signing of David Moyes wouldn’t have just happened over night.
“It would have been happening over the last couple of weeks, maybe a month that they would have made advances towards him.
“So he would have looked at the staff that he’s got in terms of the players and he knows probably how he wants to play straight away.
“Where they have got confused this season is that they’ve changed their system so many times West Ham in trying to find the right formation, trying to find the right personnel.
“I think David will try to sit into that quickly, get a formation, stick by it, and try to get the results that they need.
“Try to get the players that he’s got that can change games, like (Manuel) Lanzini, like (Michail) Antonio, on side and playing well.
“They’re the difference.”