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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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.