The report states that the the Serie A duo will make a move for the 27-year-old before the end of the summer transfer window.
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.
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.
Lucas made 32 appearances for Liverpool in all competitions last season.
The Black Eagles won their opening game of the season 5-2 against Mersin on Sunday 16 August.