ESPN reports that Moyes is on the verge of taking the Hammers job, and there had been talk that Neville - who played under Moyes at Everton and worked as part of his coaching team at Manchester United - may join him at the London Stadium.
The Black Cats are in a desperate situation in the Championship. They go into the international break bottom of the second tier just months after being relegated from the Premier League - and parted company with former manager Simon Grayson after their 3-3 draw with Bolton last week.
While the report does not suggest how interested Sunderland are in appointing Neville at this stage, the fact that he has seemingly rejected Moyes' offer to join him at West Ham certainly hints that the retired defender is somewhat confident about his chances of landing the Stadium of Light post.
But with Sunderland in a terrible position, it would not be a surprise if Black Cats supporters were less than enthused about the prospect of Neville becoming the man tasked with getting them out of their current situation.