The German is in the second of a two season loan with the Turks, though now wants the deal immediately scrapped.
The claim was made by Besiktas board member Erdal Torunogullari during an Instagram live broadcast on Monday evening.
"Karius has caused us trouble," said Torunogullari, as reported by A Spor in Turkey. "He has gone to FIFA over his unpaid wages and termination of his contract.
"He has asked for something he doesn't deserve. He want the money from the months where he hasn't been playing.
"He wants to leave, and that's his own business. But we do not want to make an unfair payment."