Karius, 26, joined the Turkish club on a two-year loan from Liverpool in August 2018, a deal that is nearing its conclusion.
The goalkeeper is contracted with the Premier League giants until 2022 and said he maintained contact with Klopp.
"I primarily write with goalkeeper coach John Achterberg. Almost weekly, he is my first point of contact," Karius told Sport Bild on Wednesday.
"But now and then I also write with Jurgen Klopp, I am in good contact with everyone. I was never out of it."
Hertha Berlin have been linked with a permanent move for Karius, whose last competitive appearance for Liverpool was his calamitous performance against Real Madrid in the 2018 Champions League final.
Karius refused to be drawn on his future, saying there was even greater uncertainty due to the coronavirus pandemic.
"I am not involved in speculation. The fact is that I have a contract in Liverpool until 2022," he said.
"And at the moment I'm just concentrating on the season at Besiktas. It is far too early to say anything about summer, especially now that nobody knows exactly what to do because of corona."