The Toffees have been struggling with injuries so far this season.
Moyes told the Daily Mirror: “We can push to be in the top six.
“We haven’t got an awful lot in terms of player numbers but what we have got is a real determination to continue succeeding at our level.
“I am going to work the players hard and I have told them I expect a top six finish from them.”
He added: “I still think after looking at last week’s game against Villa and the Premier League as a whole we have got a chance for the top six, even with the injuries we have.
“I think our performance against Villa proved we can still push to be in the top six.”