The 33-year-old left-back, who will sign a two-year contract, is manager Neil Warnock's second summer signing following the arrival last month of Jason Pearce from Portsmouth.
"He's a very good professional," Warnock told the club's official website. "I think he's a good defender, he's quick and I thought we needed a bit of experience in certain places, and that desire and will. And with Adam you know what you're going to get.
"I'm delighted to have him. He's got the experience we need. He played 16 times for Norwich last season and he's better than a lot of full backs in the Premier League.
"He's a good lad. That's half the battle as well. It gives us a good mix because we've got young Pearcey on that side as well so we can put that to bed now and move on to a different area. When you speak to him he's full of enthusiasm and when you get released from a club you want to prove people wrong."
Drury, awarded a testimonial match against Celtic at Carrow Road in May, has made over 350 appearances for Norwich, who he joined from Peterborough for £500,00 in March 2001.