News the deal had gone through came courtesy of the team's official Twitter account:
Christian Falk of Sport Bild posted the following snap of the stopper on Saturday afternoon seemingly flying out to Turkey for his medical.
Karius departs Anfield following the signing of Alisson Becker from Roma in the summer, with the Brazilian arriving in what was then a world-record transfer.
According to Sky Sports, Besiktas will pay £1.25 million for each year of the loan transfer, as well as another £1.79 million when the temporary move comes to an end.
Karius arrived at Liverpool in 2016 from Mainz with a reputation as one of the best young goalkeepers in Europe, although he never convinced when handed a starting berth.
Last term he was made first choice by manager Jurgen Klopp and played a big role in the Reds reaching the UEFA Champions League final. However, the German made two massive errors in the final that eventually helped Real Madrid to a 3-1 win.
OptaJoe commented on what was a disappointing end to the 2017-18 campaign for the 25-year-old:
After making such high-profile errors there were question marks as to whether Karius would be able to bounce back or whether he was good enough to be a starting goalkeeper for the Reds. The signing of Alisson was an emphatic indicator that he doesn't think so.
Karius should benefit from getting away from Liverpool after such a catastrophic performance in the Champions League final and will hope to regain some confidence in Turkey. If he can do so, the goalkeeper has the talent to rebuild his career and help the Super Lig giants this term.