Per the source, the Reds are open to a loan move with the view that Benteke could rediscover his form as well as gain valuable Champions League experience.
Mario Gomez ended last season as top goal scorer, helped Besiktas win the league title and earned a recall to the Germany national team ahead of Euro 2016 this summer where he scored twice.
Demba Ba also rekindled his fledgling career at Besiktas before earning himself a big money move to China.
Besiktas have developed a reputation for developing talent and turning round the careers of underperforming experienced former stars.
The Echo report that Liverpool want £32.5 million for the Belgium international striker and that Jurgen Klopp will not consider offers below the asking price.
Besiktas however, want a temporary move which the Reds may consider. The Istanbul giants do however want a future buyout clause option if they are to loan Benteke but could find a compromise with the Reds if talks go to plan.
The Belgium international is reported to be open to the move with the prospect of playing regular first team football for a side that will challenge for the league title and play guaranteed Champions League football.