Leiva is keen on playing in Turkey but the clubs are yet to reach an agreement.
Bulut told Fanatik: “Talks are ongoing, Leiva wants to play for Besiktas and an agreement should be reached between the clubs soon.”
Bulut is the agent of Arda Turan and has close ties to Gestifute headed by super-agent Jorge Mendes. Bulut also played a key role in the Black Eagles signing Mario Gomez last month.
Leiva is expected to leave the Reds over the transfer window after falling out of favour with Brendan Rodgers.
NTV Spor reported today that Besiktas president Fikret Orman traveled to Liverpool to complete the transfer of Leiva.
The Istanbul giants are in need of a central midfielder following injuries to Veli Kavlak and Tolgay Arslan who are both ruled out until the turn of the year.
Series A giants Juventus and Inter Milan are also reported to be preparing a move for the Brazilian midfielder according to Eurosport.
The 28-year-old midfielder made 32 appearances for the Reds in all competitions last season despite a series of injury problems.
The Black Eagles won their opening game of the season 5-2 against Mersin and take on Trabzonspor next at the Ataturk Olympic stadium.