Nash is available on a free transfer after being released by Stoke earlier this summer.
The 39-year-old has a wealth of experience making his name at Stockport County before joining Manchester City in 2001.
Norwich boss Chris Hughton is in the market for goalkeeping reinforcements having seen Jed Steer move to Aston Villa and also Lee Camp depart following the expiration of his short-term deal at Carrow Road.
The Canaries are also fighting to keep hold of John Ruddy, with Chelsea keen on the England keeper and Nash would prove a capable deputy to Mark Bunn and Ruddy.
However, they could face competition from Cardiff for Nash with Malky Mackay in the market for a No.3 goalkeeper to provide cover for David Marshall and Joe Lewis.