The Liverpool goalkeeper was in the second year of his time at the Turkish side - where he had made 30 appearances before the coronavirus pandemic halted the season - but he has now ended his run after a pay dispute.
In a lengthy Instagram statement, he claimed: "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.
"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."
The 26-year-old shot stopper added that he "really enjoyed playing" in Istanbul, and he thanked the fans for their "amazing support", as well as his teammates and the staff for their commitment.
He continued: "It's important to me that you know I really enjoyed playing for this club a lot.
"Besiktas can be proud of having such passionate fans behind them always giving amazing support. You always supported me in good and bad times and I will always remember you in the best way!
"Also I want to say thank you to all my teammates, coaching staff including all people working for the club.
"You welcome me with arms wide open from day one. Thank you so much!"