After their 3-2 defeat at Hull City last night, Moyes delivered a bad-tempered post-match media conference.
One question concerned the club's position in the table, 14th, four points above the bottom three.
Moyes was asked if he was worried Everton could become embroiled in a survival battle, and he replied: "Could be, yes."
Moyes would also not comment on news filtering through tonight that the club have failed to secure permission to build a new stadium.
He said: "We didn't play well enough to win, that sums it up really.
"I don't think it was anything to do with confidence, it was to do with endeavour and attitude, which Hull City showed.
"If we had matched Hull's endeavour and attitude - they ran around and it earned them something at the end of the night.
"A team should fight from the first minute, not the 45th."