Beşiktaş have yet to agree a new contract with the 26-year-old full-back who has just six months remaining on his current deal.
Liverpool and Valencia could sign Koybaşı as a free agent should the Black Eagles fail to reach an agreement with the Turkish international.
Koybaşı has established himself as a key player in the Beşiktaş side averaging 2.2 tackles and 2.8 interceptions per-game.
The pacy left-back also likes to get forward and is not afraid to take on opposition defenders averaging 1.3 successful dribbles per-game – the joint third highest for his side.
The Black Eagles are currently second in the Turkish Super League just two points behind Istanbul rivals Fenerbahçe but have a game in hand after the weekend clash against Mersin was abandoned due to heavy snow.