With wins in their last matches both the Whites and the Toffees opened a gap of seven points and three points respectively over 18th-placed Watford, but the relegation battle still remains in the balance.
The Toffees have had to face some of their stars like Dominic Calvert-Lewin and Richarlison having subpar seasons, while the Whites have had to deal with defensive disasters, and both have struggled with injuries.
Neville believes that Leeds or Everton could still go down from the top flight as he has an inkling that there is something in their squads that could hold them back from surviving.
The Manchester United legend feels Watford manager Roy Hodgson and Burnley boss Sean Dyche will be looking at both the Toffees and the Whites and thinking that they could overtake the duo.
Speaking on The Overlap, Neville said: “I think Everton or Leeds could go down.
“Leeds’ win over Wolves was massive to make it back-to-back wins, that could save them.
“But I do think there is something in both of those squads where you think they could go.
“I look at Sean Dyche and Roy Hodgson and I think they know what is going on here.
“Something could happen at Watford.
“Roy has done it before with Fulham where they managed to get out of it on the final day.”
Hornets manager Hodgson secured Fulham’s survival in the in the 2007/08 season, after similarly being appointed mid-season, and Watford won their last match as well, while Burnley won two games back-to-back just last month.
Burnley have also lost just 12 games, a total better than any of the other clubs in the bottom six.
