The 43-year-old Korkut takes over from Hannes Wolf, who was fired on Sunday after a home defeat to Schalke 04 left the newly promoted side three points above the Bundesliga relegation places.
Korkut led Hannover to a 10th place finish in 2015 before brief stints at Kaiserslautern and Bayer Leverkusen the following two seasons.
He has also served as assistant national team coach for Turkey and even played for them at Euro 96 when he was doing well as a Fenerbahce midfielder.
Stuttgart signed veteran German international striker Mario Gomez, 32, in the winter break, but his arrival has failed to halt the slide.
The burly forward, back at the club where he started his career before joining Bayern Munich in 2009, is still waiting for his first goal after three games.
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