The Red’s chief analyst Dave McDonough and scout Kevin Hunt were present at the Istanbul derby on Monday according the the Turkish newspaper AMK.
The Istanbul giants are yet to agree a new contract with the full-back who has under six months remaining on his current deal.
Liverpool could sign Koybasi on a free should the Besiktas fail to reach an agreement with the former Gaziantepspor man.
Koybasi has established himself as a key player in the Besiktas side averaging 2.2 tackles and 2.8 interceptions per-game.
The Turkish international 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.
Besiktas are currently second in the Turkish Super League just two points behind Istanbul rivals Fenerbahçe but have a game in hand.