Joining the club last summer, Onyekuru was on loan at Anderlecht, where he scored nine goals in 19 games.
He came back to Everton in January due to a knee injury that kept him out for the rest of the season.
Now he is approaching full fitness. His club are hoping to get another loan deal sealed.
And Sky Sports believes that Galatasaray are at the front of the queue.
The club mentioned on Twitter that "formal negotiations" had begun for the forward.