The report claims that Besiktas have made an enquiry to Liverpool for Leiva and that initial talks with his agent proved fruitful.
Leiva was closely linked with a move to the Black Eagles last summer but former Reds boss Brendan Rodgers blocked the transfer.
The Reds rejected a loan-bid from for Leiva last summer according to the Liverpool Echo but Besiktas are in a stronger position this season.
The Istanbul giants won the league title and are guaranteed Champions League football next season.
The Brazilian midfielder’s future at Liverpool is in doubt after he admitted concern over his future at the club next season.
“There have been some changes last season, with the coaching change,” Leiva said, speaking to Radio Joven Pan.
“I still do not know exactly what will be my role within the group, but I am returning (to England) on Thursday to start the pre-season.
“It is the first transfer window in which Klopp will participate more actively in Liverpool, so I’ll just get a better idea when I return.”
Besiktas are in pursuit of a central midfielder as it was an area the Eagles lacked depth and Atiba Hutchinson could be on his way out.