The 34 year-old has joined the Premier League champions as back-up to first-choice Joe Hart and Costel Pantilimon.
The move is one of the more surprising of the summer transfer window.
Wright, who has made 467 career appearances and won two England caps, joined League One Preston earlier in the summer but left a week later citing homesickness.
Wright made his name as a teenager with Ipswich, for whom he played more than 300 games over three spells.
He joined Arsenal in 2001 and has also played for West Ham, Southampton and Sheffield United.