Karius has ended his two-year loan deal with the club after he reported them to FIFA for allegedly not paying his wages.
"It's a shame it comes to an end like this but you should know that I have tried everything to solve this situation without any problems," Karius said on Instagram.
"I was very patient for months telling the board over and over again. Same things happened already last year. Unfortunately they haven't tried to solve this situational problem and even refused my suggestion to help by taking a pay cut.
"It's important to me that you know I really enjoyed playing for this club a lot. Besiktas can be proud having such passionate fans behind them always giving amazing support."
FIFA told ESPN it is investigating the outstanding payments in a statement: "We can confirm that we have received a claim from the player Loris Karius against the club Besiktas.
"The matter is currently being investigated and consequently we cannot provide any further comments."
The 26-year-old joined Besiktas after he made two mistakes in Liverpool's 3-1 defeat to Real Madrid in the 2018 Champions League final.
Karius has been linked with a return to the Bundesliga where he played at Mainz before joining Anfield in the summer of 2016.
Karius is still in Istanbul and is hoping to return to Germany as soon as possible. He made a total of 67 competitive appearances for Besiktas.