The 27-year-old has not played for the Reds since making a couple of errors in the 2018 Champions League final defeat to Real Madrid.
Karius spent the following season with Besiktas and then moved to Union on a 12-month loan last September, but he has struggled to oust Andreas Luthe as first choice.
However, despite making just four Bundesliga appearances this campaign, the German stopper is keen to stay on at Stadion An der Alten Forsterei beyond this summer.
"I still have a contract with Liverpool until 2022. But of course I can imagine staying in Berlin if everything fits," he told Bild.
"I'm not just passing through here. The mentality of the people and the spirit of the team are good for me. I'm basically a down-to-earth guy, even if I drive an expensive car.
"I feel really good in Berlin and at Union. Of course I want to play, but if that were different, then I wouldn't have lost anything here either."
Karius made a total of 49 appearances for Liverpool in all competitions during his two seasons on Merseyside.