Sky Sports reports that the 34-year-old is undergoing a medical at Eastlands.
City boss Roberto Mancini is keen to bring in Wright as additional back-up for undisputed No.1 Joe Hart.
Wright could sign a contract with City in the next couple of days.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.