The Toffees boss, who last experienced a final in the domestic competition in 2009, believes that the third-round fixtures have become a headache for the majority of clubs.
"The FA Cup is maybe losing a bit of its shine," the Daily Mail quotes Moyes as saying. "At the moment the Premier League is so vital.
"The money that is involved in staying in the Premier League and finishing in a good position far outweighs the need for a good cup run in so many clubs' cases.
"I wouldn't give a Champions' League place [to the FA Cup winners]. That's probably too much. But I don't know how else you do it. Maybe we need to slip the third round in [to the calendar] a little bit earlier."
Moyes's side will go head to head with fellow Premier League club Fulham in the fourth round of the FA Cup following their 2-0 win over Tamworth on Saturday.