Their late defeat at West Ham in midweek has heaped further intense pressure on the five-times European champions.
Two weeks ago Everton boss Moyes claimed his own side were best placed to win the race to qualify for the Champions League.
But he said on Friday: "Competition for fourth is very high, but I still believe Liverpool are favourites for that place, they have the game in hand on everyone and I expect them to get there.
"They would still be the favourites. But we are in there pushing, fighting for that Champions League spot.
"It is too early at the moment to talk about it, everything can change because it is so close.
"I just hope we are still in this position come the end of the season when the run-in starts. We aim to go right to the end to achieve it.