Kokcu has become an important part of Besiktas' attacking play since leaving Benfica, contributing 14 goals and 12 assists in 53 official appearances for the Turkish club.
His performances have attracted attention in England, with Portuguese and Turkish reports claiming Liverpool are considering a move for the Turkey international.
The Reds are not, however, understood to have made a formal offer at this stage.
Liverpool reportedly want to continue assessing Kokcu before deciding whether to make an approach, with scouts expected to attend his matches as the club evaluates the midfielder.
Besiktas have made their own position clear. The Turkish club do not want to sell Kokcu during the January transfer window and regard him as an important part of their plans for the season.
Their ambitions include competing for domestic honours and progressing in European competition.
Kokcu has made an immediate impact this season, registering four goals and five assists in 12 appearances for Besiktas.
The 25-year-old's familiarity with European football and his ability to contribute both creatively and in front of goal are among the qualities attracting interest.
Liverpool's midfield has undergone significant changes in recent seasons, and the club continue to assess potential options as they plan for future transfer windows.
Any move for Kokcu would therefore depend not only on Liverpool's assessment of his performances but also on Besiktas' willingness to negotiate.
There is another developing situation at Liverpool to monitor during the international break.
Alexander Isak has returned to his club after sustaining a minor injury while on international duty with Sweden.
The 27-year-old played the full 90 minutes and scored in Sweden's 2-1 Nations League victory over Romania, but he will now miss the remaining three matches of the international break.
Liverpool's medical staff will assess the striker following his return to the club.
Isak has made an impressive start to the Premier League season, scoring four goals in his first five appearances for the Reds.
His debut season at Liverpool following his £125 million move from Newcastle United was disrupted by injury, restricting him to 14 Premier League appearances and three goals.
Liverpool will therefore be hoping that the latest problem proves minor, particularly given Isak's importance to the team's attack.
At the same time, the club's continued monitoring of Kokcu illustrates that recruitment work is continuing alongside the immediate demands of the season.