The Toffees are fourth in the table with eight matches gone after Sunday's 1-1 draw at Queens Park Rangers. Moyes guided Everton to the qualifying rounds of Europe's elite competition in 2005 and believes another top-four finish is within reach of his current crop of players. But the Scot, whose side have been notoriously slow starters in recent seasons, is fully aware of the challenges ahead.
"I think we have a good team and good players. I think we can try and do it [finish fourth]," said Moyes.
"It is going to be hard with the competition we have got and who knows, we might dwindle away.
"But in recent seasons the question probably 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, who have lost just once in the top flight this season, head into next weekend's Merseyside derby on the back of a five-match unbeaten streak. The Toffees host neighbours Liverpool at Goodison Park on Sunday and Moyes is feeling confident.
"We are in a healthier position than maybe we have been in for a long time before," Moyes added.
"We will go into it with a bit of momentum."