The Turk replaces Roger Schmidt, who was sacked following Saturday's heavy 6-2 defeat to Bundesliga rivals Borussia Dortmund.
Korkut, a former Turkey midfielder, resigned as boss of second division side Kaiserslautern in December after just six months in charge, having previously been in charge of Hannover.
Leverkusen managing director Michael Schade said on the club's official website: "In Tayfun Korkut we have got a coach who has a great deal of expertise and has gained a lot of experience both as a coach and a player on the international stage.
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"With him we want to overcome the difficult period we are experiencing and end the season well."
Korkut said he was looking forward "to working with very talented players at a top club".
Leverkusen have dropped to 10th place in the Bundesliga table, 11 points off the Champions League places, and have lost their last three games in all competitions.
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