The report also claims that Torunoğulları will hold talks with Liverpool over the transfer of Lucas Leiva.
The Black Eagles want Leiva on loan and the Reds were reportedly interested in letting the Brazilian leave until injury to Jordan Henderson.
Leiva started his first game of the season for Liverpool in the goalless draw against Arsenal last Monday at the Emirates.
The Liverpool Echo reported that the Reds rejected the loan-bid from Beşiktaş for Leiva following the game.
The Black Eagles are in the market for a central midfielder following injury to Veli Kavlak and Tolgay Arslan – both are ruled out until the turn of the year.
The Istanbul giants are currently first in the Turkish Super league with six points after three games and a goal average of six.