The Belgian striker has struggled for his best form at Anfield.
Sporx are reporting that the Turkish side are keen on bringing the Belgian striker on loan in the summer.
Fenerbahce managed to pluck Serbian Lazar Markovic away from Liverpool on a similar deal this summer - so are hopeful of pulling off a similar deal for Benteke.
It would not come as a surprise.
Since making his big money move to the club in the summer from Aston Villa the striker has struggled to find his best form at Anfield.
He cost a fee reported by the Guardian to be in the region of £32.5 million but he has struggled to live up to that price-tag on Merseyside.
He has netted just seven goals for the club and remains goalless in 2016.
Benteke would also not seem to fit into the playing philosophy of Liverpool boss Jurgen Klopp - who asks for high energy play from the front to the back in his team.
So an exit certainly appears likely for Benteke and it looks like Fenerbahce are ready to make their move.