The Turkish side do not want to pay a free for the striker, who is expected to leave Anfield this summer.
The striker is expected to leave Anfield this summer after struggling since joining from Aston Villa last summer.
He has failed to impress either former manager Brendan Rodgers or his successor Jurgen Klopp, with the Mirror reporting that the current Liverpool boss has told him he is free to find himself a new club this summer.
Christian Benteke scores the first goal for Liverpool
Crystal Palace had appeared to be his most likely destination, but so far the two sides have failed to agree a move as the Reds look to recoup the £30 million they initially paid for his services.
That leaves the door open for other clubs to make a move and it appears that Besiktas have tried their luck.
According to NTVSpor, they approached Liverpool with a loan offer but that was quickly rejected as they are looking for a permanent exit.
However the Turkish side do want to pay a fee for Benteke and as a result will return with another offer to try to tempt the Reds.
It remains to be seen if the striker is interested in the move too, with him surely hoping for a Premier League club to come in and rescue him from his Anfield nightmare.