The report states that the Black Eagles have made an enquiry through football agent Ahmet Bulut who will lead talks.
The agent also represents Arda Turan and has close ties to the Gestifute football agency company headed by Jorge Mendes.
Bulut also played a key role in bringing Germany international Mario Gomez to Beşiktaş over the summer.
The Brazilian midfielder has fallen down the pecking order at Anfield and could end up leaving over the transfer window.
Lucas lost his place as Brendan Rodgers deep-lying midfielder after an injury plagued season.
The experienced midfielder did not feature in the Reds opening day win against Stoke City at the Britannia stadium.
“[Rodgers] believes Lucas is not dynamic enough and prefers Joe Allen in the midfield holding role,” wrote the Mirror‘s David Anderson.
“The Anfield boss feels he would have enough central midfield options in [James] Milner, Allen, Jordan Henderson and Emre Can if Lucas went.”
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.