The Turkish club have been in talks with the Reds over a potential two-year loan deal for Karius, and appear to have completed the deal, posting a picture of the goalkeeper in their home shirt on their social media account.
Reds manager Jurgen Klopp claimed over the weekend that he was not aware of interest in the goalkeeper, maintaining that he was not told of a potential transfer.
However, it appears that Karius is now poised to rubber-stamp his move to the Turkish giants.
Liverpool signed Alisson Becker over the summer in a deal that made him the most expensive goalkeeper in world football, while Simon Mignolet will provide support to the former Roma stopper.
Karius was on the bench in the Reds' opening game of the season, as they beat West Ham 4-0, but he was not included in the squad for their 2-0 win over Crystal Palace.
He has made a total of 29 appearances for the club, but has also made a number of high-profile errors, most notably in the Champions League final, where he made two major mistakes as Real Madrid beat the Reds 3-1.
He will replace Fabri at Besiktas, who joined newly-promoted Fulham earlier in the transfer window as he aims to resurrect his career away from the prying eyes of the Premier League.