The Black Eagles could however, face competition from Inter and Juventus who have also been linked with the Brazilian.
The Istanbul giants have already made an enquiry for the experienced midfielder who is expected to leave the Reds before the end of the transfer window.
Lucas lost his place in the Liverpool team after an injury plagued season. The experienced midfielder did not feature in the Reds opening day win against Stoke City at the Britannia stadium or against Bournemouth at Anfield.
Beşiktaş are currently suffering an injury crisis in central midfield. Tolgay Arslan and Veli Kavlak are both out injured until the turn of the year.
The Black Eagles won their opening game of the season 5-2 against Mersin on Sunday 16 August.
Lucas joined Liverpool in 2007 and has 22 international caps for the Brazil national team.