The Reds released a statement confirming the move, after Besiktas' social media team appeared to jump the gun by posting an image of the German stopper on Twitter on Friday.
Karius, who joined the Reds two years ago, appeared to have ousted Simon Mignolet for the number one spot last season. However, costly errors during the Champions League final loss to Real Madrid at the end of the campaign drew question marks over his future.
While Jurgen Klopp rejected claims that the performance caused him to search for a new stopper, the Reds brought in Brazilian star Alisson Becker from AS Roma for £67m, pushing Karius out of the first team picture.
A statement on Liverpool's website reads: "
"The 25-year-old is set to spend the remainder of 2018-19 and the whole of 2019-20 with the Super Lig outfit, who have started the new campaign with back-to-back league victories.
Karius's move to Besiktas was stalled by Turkish holidays, but he has officially left Anfield, with the clubs having agreed over a two-year, £2.25m loan move, per reports which also claim that Besiktas will be obligated to buy him outright for an additional £7.25m as long as certain criteria are met.