The German goalkeeper started May's Champions League final but made two high-profile errors - though tests later revealed he had suffered a concussion during the game – while Karius was guilty of further blunders in two of Liverpool's pre-season friendlies.
With Simon Mignolet now seemingly set to back up new signing Alisson, Karius may be offloaded just three months after he was Klopp's first-choice keeper.
Besiktas appeared to announce the move on social media Friday, but Klopp had nothing to add about the potential transfer.
"There's nothing new about Loris' actual situation," Klopp told a news conference.
"Nothing's decided. If it happens, that's how things go in football. That can happen.
"As a manager you usually don't need that much time between one game and the other game to get the sack. But it's not the sack, it's only a change if it's like that.
"That's football. No problem with that. Whatever. Things in football change. If they change, then we speak about that. Before that, I have nothing to say about that."
Liverpool have won their opening two games of the Premier League season with the same starting line-up and face Brighton and Hove Albion at Anfield on Saturday.
Klopp confirmed Croatia international Dejan Lovren will once more miss out with a pelvis issue that is likely to keep him sidelined until after the international break.
"He's coming closer but he's not really around team training at the moment," Klopp said.
"I am not 100 per cent what he is doing today. He will not be involved in that [Brighton] game, and I am pretty sure he has no chance for the Leicester [City] game.
"Then hopefully after the international break he will be ready to go, that would be nice. That was the target from the first day when he was in and we knew about his problems. We always thought if we could give him that time, then that would be cool."
Klopp also praised Sadio Mane's development, with the ex-Southampton forward having scored three times already this term.
"It’s his third season here and he’s constantly improved," Klopp added.
"We knew we would get a good player when we signed him from Southampton. Did we exactly know how good? I wouldn’t say so.
"You sign a player and then you hope he could make a few more steps and that’s what he did."
Asked about whether there was an update on the Senegal international's contract situation, Klopp replied: "Yes. He has still a very long contract [until 2021]! That’s not new but that's a fact."