The 46-year-old German spent four years with the Canaries and twice led them to promotion to the Premier League.
Farke spent last season with Bundesliga side Borussia Monchengladbach but was sacked after they finished 10th.
Leeds were relegated from the Premier League in 2022-23 after a campaign that saw them sack two managers, with Sam Allardyce then unable to keep them up.
“I feel humble at this time, I know the responsibility to fulfil all the expectations and I want to repay the trust shown,” Farke told the club website.
“The most important thing is to create a togetherness and unity within this club again and, from today onwards, I will work on it with my staff and players, and I trust our supporters will be there when we need them. I can’t wait for the first game of the season.”
Farke was appointed manager at Carrow Road in June 2017 after a spell as head coach of Borussia Dortmund reserves, and guided the Canaries to the second-tier title in his second full season in charge.
They finished bottom of the Premier League the following campaign but Farke led them straight back up, once again as champions.
After a poor start to the 2021-22 campaign he was sacked in November – hours after their first win of the top-flight season, at Brentford.
Farke subsequently had a brief spell in charge of Russian side Krasnodar, but left without managing a game because of Russia’s invasion of Ukraine, and then took over at Monchengladbach on a three-year deal last summer.
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