However, Lucas, who has two years to run on his contract, returned to the starting line-up for Monday's goalless draw with Arsenal following an injury to skipper Jordan Henderson and Rodgers praised his performance against the Gunners.
"Lucas is our best defensive midfield player," Rodgers said after the Arsenal game.
"There are games where I play three midfield players but not always with a defensive specific one, but when we do need that, he is the best one we have," the manager added.
"He's certainly very important to what we want to do."
Besiktas, who have been hunting for midfield reinforcements after losing Tolgay Arslan and Veli Kavlak to injury, have confirmed that Liverpool are now unwilling to sell the Brazilian.
"Liverpool do not want him to go. The transfer can only go ahead with their say so there is nothing we can do right now," the Echo quoted Besiktas president Fikret Orman as saying.
The midfielder is reportedly training with the Liverpool squad ahead of West Ham United's visit to Anfield on Saturday.
(Reporting by Simon Jennings in Bengaluru; editing by Amlan Chakraborty)
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