Torunoğulları was quoted by Lig Tv as saying: “Lucas is a player on our shortlist, talks are ongoing.”
The Lig Tv report claims that the Black Eagles want the Brazilian midfielder on a loan deal after the Reds requested a €7 million transfer fee.
Leiva has been left out of both Liverpool games this season despite recovering from injury and is expected to part ways with the club according to the Guardian.
The 28-year-old midfielder still has another two-years left on his contract and has made a total of 275 appearances for the Reds since joining on a £7 million move from Gremio in 2007.
Beşiktaş are in the transfer market for a central midfielder following injury to Toglay Arslan and Veli Kavlak. Both players have been ruled out until the turn of the year.
The Black Eagles finished last season in third place and will compete in the Europa League group stages this season.