The Croatian striker has struggled during his second season at Goodison Park, scoring eight goals in 39 appearances.
That return comes after the 27-year-old netted 11 times in the final 16 games of the last campaign following his move from Rangers.
"His effort has never been in question and we want him here next year," Moyes is quoted as saying by The Sun. "It's been a difficult season — maybe a bit of second-season syndrome.
"He's found it a little harder to get the goals and the opportunities."
Jelavic has scored once in his last 11 games for Everton in all competitions.