The Reds will recoup £2.25 million over the period of the initial deal, while a further £7.25m will be due if he hits certain performance criteria.
Should these targets be hit, Karius' move will automatically become permanent. It is understood these are primarily based around appearances and relies on his club reaching European football.
Meanwhile, the German will sign a one-year extension with Liverpool should he return to Anfield, while he will have the option of another season after that, potentially taking him through to 2022.
Besiktas' goalkeeper hunt began when they lost Fabri to Fulham, which was a move triggered by Simon Mignolet's refusal to move to the London side.
Senol Gunes's club, meanwhile, had been keen on signing the Belgian from Liverpool, but Goal understands Besiktas ended their interest due to a high transfer fee and the player's wage demands.
That means Belgian Mignolet is set to remain at the club and act as back-up for summer signing Alisson, with Kamil Grabara third choice.
Liverpool will not stand in Karius's way as he seeks regular first-team football having suffered a difficult last few months following his high-profile errors in the Champions League final.
Their preferred option was always to send him out on loan, unless they received a market-value offer for Mignolet, which never materialised.
Jurgen Klopp had given the German goalkeeper game-time during pre-season, while he was given an excellent reception by Reds fans as he appeared at Anfield in a friendly win over Torino at the beginning of August.
Despite the likelihood being that Karius leaves Liverpool before the transfer window closes for the rest of Europe, he is still with the travelling party for Monday's Premier League clash with Crystal Palace.
The Reds will be looking to build on their impressive opening-weekend victory over West Ham as they face former manager Roy Hodgson - who himself led Palace to a win in their first game as they beat Fulham 2-0.
Karius has made 49 appearances in all competitions for Liverpool up to now, including 19 in the Premier League last season.