It is believed that talks are underway between former Liverpool director Damien Comolli - who is now Fener's sporting director - and Everton over a move that would see the 29-year-old move to Turkey.
The report says that DR Congo international Bolasie was unsure over a move to move to Turkey with Fenerbahce, but 27-year-old striker Cenk Tosun has appeared to persuade the former Crystal Palace star to give it a try.
Bolasie joined Everton in August 2016, but suffered a long-term knee injury just a few months into his time at Goodison Park. Finally returning to the Everton side in December 2017, Bolasie made 17 appearances in all competitions.
But the ex-Bristol City trickster failed to replicate the kind of form that saw him earn a move to Everton in the first place, and now he could be loaned to Fenerbahce.
Bolasie still has three years remaining on his five-year contract at Everton, which the Daily Star claim is worth £75,000 a week. If he does join Fener though, he may leave permanently before the expiration of his contract.