The Merseyside outfit currently sit in fourth position in the league table following eight games of the campaign.
"I think we have got a good team and good players, I think we can try and [finish fourth]," the Daily Mail quotes Moyes as saying. "It is going to be hard with the competition we have got and who knows, we might dwindle away.
"But in recent seasons probably the question that is being asked is can we avoid relegation, so it is nice to be up there and I'm going to be positive about it."
Everton clinched a point over the weekend courtesy of a 1-1 draw with Queens Park Rangers.