The 42-year-old former Turkey international, who coached Hanover from early 2014 to 2015, was more recently coach of second-tier Kaiserslautern until December.
"I am confident we can finish the season as Leverkusen had imagined it," Korkut told reporters. "I am happy to take on this task and work with such a talented team."
Leverkusen have managed just three wins in their last nine league games in an erratic campaign, and they are 13 points away from the third place that leads to the Champions League group stage.
They travel to Atletico Madrid on March 15 for the return leg of their Champions League round-of-16 match after a 4-2 loss at home in the first leg last month.
"We can still clinch a European spot, that's achievable. I am optimistic with our quality we can get back on track," said Leverkusen sports director Rudi Voeller, who said a contract extension was also possible for Korkut.
"This is a chance for him and for us. I know he is extremely motivated and ready to go. Everything is possible when we talk in the coming weeks, months or after the end of the season," Voeller said.
Leverkusen also suffered an ignominious German Cup second-round exit at the hands of third-tier club Lotte this season.
(Reporting by Karolos Grohmann, editing by Larry King)
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