Former Liverpool defender Carragher was asked which team he believed would succumb to the drop during Sky's Monday Night Football broadcast, which saw Newcastle United defeat Arsenal 2-0.
Carragher said: "I think Everton will be okay. I do believe that. Listen, there'll be a real panic if they don't get a good result against Crystal Palace but Everton have got to remember where they were two weeks ago against Chelsea when they were, I think, five points behind, so they're still in a really good position. I fear for Leeds. I think Burnley can get something out of their next two games.
"I'd still say Leeds go down."
Ex-Man United full-back Neville was in agreement with his television colleague, adding: "I think Leeds [will be relegated], I think Burnley get something, I don't think Leeds get something at Brentford. I might be wrong but I think Leeds are in trouble."
Whites supporters will be hoping for Neville's prediction to be proved wrong this Sunday when they take on Brentford in west London.
The goalposts could shift quite considerably on Thursday evening when both Burnley and Everton play their games in hand.
Positive results for one or both would make United's task a great deal harder at the weekend, although defeats for the Toffees and Clarets would give Leeds a survival boost heading into the final round of fixtures.
Following Newcastle's victory over Champions League-chasing Arsenal, Neville opined: "We need a strong Newcastle, we need a strong Sunderland, Middlesbrough, we need a strong Leeds United."
Neville insinuated that some northern clubs had long been neglected by ownership and that teams such as those listed performing well, made for a better English game.