Karius joined the Turkish club on a two-year loan on Sunday after losing his No 1 spot at Anfield to summer signing Alisson.
Karius believes his loan spell in Turkey will be good for all parties involved, but says he had to convince Klopp to let him go.
Speaking at his Besiktas unveiling, Karius said: “He [Klopp] told me he would like to have me in the squad because he wants to have the best quality players.
“But I spoke to him and asked him for my wish, it is important to me to play for Besiktas. I talked with him for a long time and he understood my situation.
“I think it was a good deal for everyone. He wished me well but he said he would have liked to keep me.”
Karius is determined to help Besiktas win the Turkish Super Lig title after a disappointing fourth-place finish last season.
“I come here because I think I can make the next step in my career, improve as a player and also help the whole team to improve,” he said.
“I’m still young for a goalkeeper. I want to make the next step and hopefully win many titles with Besiktas.”