Kelleher warmed up Allisson and Simon Mignolet before Liverpool's 1-0 win over Brighton on Saturday and the Liverpool Echo reports that Klopp is "blown away" by his development and has made him number three at Anfield.
However, the departure of Danny Ward to Leicester City and strong doubts over the future of Simon Mignolet suggested that Karius would remain as the club's back-up option.
The move could mean the exit door is closed for Mignolet, but the Belgian still wants to leave if he cannot play at Anfield.
I haven't been told anything since Loris left.
Karius's place at Liverpool had been in severe doubt following his two horrendous errors in their Champions League final defeat to Real Madrid in May. "Playing minutes is the most important thing", he added.
Speaking to Het Laatste Nieuws, he said: "I find it weird that Karius was loaned while I also had options to be loaned".
"I find it weird that Karius has been loaned, while I also had options to be loaned".
Liverpool FC will now switch their attentions towards preparing for their clash against Leicester City in the Premier League on Saturday lunchtime before the worldwide break.
And Mignolet is shocked that Karius has been loaned out given he was above him in the pecking order. We shall see what happens this week. 'But for some reason that was not possible.
A keeper who was made first choice ahead of me a year ago gets to leave on loan.
"Proud to be part of the famous Besiktas family now", Karius wrote on Twitter.More news: Playwright Neil Simon dies at 91
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