Liverpool agreed a two-year loan deal for the goalkeeper in which Besiktas are obligated to make the move permanent and trigger a £9.5m deal if Karius achieves enough first team appearances.
However, news has surfaced that Besiktas are unhappy with the player and want to explore the possibility of taking Origi instead.
The Turkish side are hopeful of signing Origi under the exact same terms as Karius.
It is thought that Karius needs to make more than 50% of appearances at the club to become permanent.
In the case the deal becomes permanent, a 20% sell-on clause will be active if Besiktas decide to move the goalkeeper on in the future.