The 33-year-old has a contract at Carrow Road until the end of June but with new terms not on the table he has decided to drop to the Championship and will move to Elland Road on July 1 after signing a two-and-a-half-year deal.
The defender has found his versatility a useful attribute in the Premier League this past year, appearing at centre-back as well as on the left of the back four, but after playing 362 matches for the Canaries he is moving on.
Norwich manager Chris Hughton, speaking to the club’s official website, wished the defender all the best: “I would like to wish Adam all the best for the coming season and in his future career.
"He has been an outstanding servant to Norwich City, and I was very aware of what he had done for the club before I arrived here.
"For a player to be at a club for 11 years, and to play so many games in that time is a very rare thing and a truly outstanding achievement.”
Drury joined the club in 2001 from Peterborough United, and was the only member of the team this season to have been in the squad when Norwich were last in the Premier League in the 2004-05 season.
The defender remained with the club when they were relegated to League One in the 2008-09 season and played his final match for Norwich in his testimonial against Celtic in May.