Karius muscled his way ahead of Simon Mignolet in the pecking order during the last campaign, and was Jurgen Klopp's first choice as the Reds battled to the Champions League final.
But two big mistakes against Real Madrid drew unwanted attention to the German, who also suffered concussion during that game.
Not content to sit on the bench, Karius will try his luck at Besiktas, who took the unusual step of 'announcing' the transfer by changing their Twitter header to a photo of the keeper.
The loan move will nevertheless be confirmed when the player lands in Istanbul, having been pictured taking off from Manchester Airport on Saturday accompanied by his family.
Also present was agent Florian Goll, a close collaborator throughout Karius' career and the man who manages his affairs along with the goalkeeper's father.
Karius has also signed a contract extension with Liverpool, where he has played 49 games to date since moving from Mainz in 2016.