Wright, a free agent after a brief spell with Preston North End, will act as back-up for Joe Hart and Costel Pantilimon.
Wright began his career with Ipswich Town and made more than 300 appearances for the Tractor Boys during three separate spells with the club.
After six years at Portman Road, Wright left Suffolk to join Arsenal in 2001 and won two England caps during his career, one against Malta and the other against Holland. He also collected 15 Under-21 caps during a spell when he was competing with City’s Nicky Weaver for international honours.
Wright left Arsenal in 2002 and joined Everton, where he remained for five years before having further spells at Ipswich (twice), Southampton, Sheffield United and Preston.
Now 34, Wright has made a total of 467 appearances to date.