Florian Goll says the Turkish club have not paid Karius, on loan from Liverpool, his full contracted amount between October 2019 and March this year.
The German has now taken legal action with FIFA, the club's president confirmed this week.
Besiktas board member Erdal Torunogullari stated: "Karius has gone to FIFA over his unpaid wages and termination of his contract.
"He wants us to pay him for the months he hasn't played. He wants to leave and that's his decision. We don't want to make an undeserved payment."
In reply, Goll said: "I strongly condemn the fact that club representatives now want to portray Loris as a bogeyman. The payments have absolutely nothing to do with the corona crisis, even if the club likes to communicate it that way."
However, he added: "We did not separate. Loris is still under contract with Besiktas."