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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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.