Although Andreas Luthe was one of the better performances in Union’s 1-1 draw against Borussia Monchengladbach, the Berlin club still want another shot-stopper.
Karius spent the last two seasons on loan at Besiktas, but a dispute over unpaid wages cut the move short in May.
The German fell out of favour at Liverpool following his mistakes against Real Madrid in the 2018 Champions League final.
The Kopites would go on to break the world record for a goalkeeper by signing Alisson from Roma.
In search of game time, Karius will spend the season in his homeland with Union Berlin.