The Barclays Premier club have confirmed 33-year-old Drury, who joined Norwich in 2001 and stayed with them during their plunge into League One and their triumphant return last year to the Premier League, would be leaving Carrow Road.
He made a total of 362 appearances for the club, scoring four goals.
Norwich City boss Chris Hughton told Canaries.co.uk: "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."
The club played a testimonial for Drury against Celtic last month.