The 39-year-old joins the club on a free transfer following his release from Stoke in the summer.
Nash will provide additional back-up to John Ruddy and Mark Bunn after Jed Steer left the club and Declan Rudd joined Preston on loan for the season.
The Canaries are Nash's 11th club of a career spanning over 20 years, including spells at Stockport, Preston, Manchester City and Everton.
Norwich boss Chris Hughton said: "It is now important to us to bring in a 'keeper of great experience and a player that is used to living life in the Premier League.
"The move also allows us to send Declan Rudd out on an extremely valuable season-long loan to Preston where he can get some first-team football under his belt."