The Black Eagles have been closely linked with the 27-year-old but were reported to be put off by his €7 million asking price. The Istanbul giants will try to loan the Brazilian international instead.
Besiktas are in need of a central midfielder following injuries to Veli Kavlak and Tolgay Arslan. Both players have been ruled out until the turn of the year due to injury.
Inter Milan and Juventus are both in the race to sign the out-of-favour defensive midfielder according to Eurosport.
Lucas lost his place as Brendan Rodgers deep-lying midfielder after an injury plagued season and is expected to leave over the transfer window.
Besiktas won their opening game of the season 5-2 against Mersin and are currently coached by former Turkish national team manager Senol Gunes who guided the Crescent-Stars to third in the World Cup.