Per the source, Besiktas president Fikret Orman made Liverpool an enquiry and was told the Reds would be open to a loan deal.
Besiktas are in pursuit of a new striker after Germany international Mario Gomez revealed that he will not be returning to the club this summer.
Gomez was top scorer for the Istanbul giants last season and played a key role in Besiktas winning the league title. The experienced striker earned a place back in the Germany national team as a result of his form last season.
Liverpool have already rejected offers for Benteke over the summer and Reds head coach Jurgen Klopp revealed that any potential suitors will have to meet their asking price if they want the Belgium international.
Klopp was quoted by the Guardian as saying: “Christian already said it maybe makes sense for him to go to another club and that’s the situation at this moment.
“Christian is still a great striker and that’s how it is, but we have a few more strikers and in this moment it makes more sense to give them match time and that is what we did.
“If there are clubs that want a striker from us they have to make real offers and are not hoping for anything else, otherwise we will find solutions.”
The Reds want £32.5 million for the the powerful striker but have yet to receive an offer matching their valuation.
Besiktas would not be able to match Liverpool’s asking price but Klopp may be prepared to let him leave on a temporary move and gain Champions League football experience this season.