Moyes led West Ham into Europe during his time in charge and won the Europa Conference League.
"It was amazing for the club and it transformed West Ham," Moyes told BBC Radio 5 Live. "European tours for the supporters, the money that was getting brought in from the home games and we got to the semi-final of the Europa League.
"I don't really see why we should not attempt to do that here at Everton. That has to be the challenge."
Moyes added: "We are seeing the odd team now maybe having a go such as Bournemouth and Fulham, while Brentford have tried to show over the years that they can get close to it.
"I don't see why we shouldn't try to get ourselves into that group. If we keep playing the way we have been doing, get results, then we will give ourselves every chance."