The Daily Mail says City will pay £26m for the Italy goalkeeper, who will be one of three transfers City conducted in the final minutes of Monday's transfer deadline.
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City keeper Ederson has been sold to Fenerbahce for £12m, while defender Manuel Akanji is joining Inter Milan on-loan for the season. Serie A are already announcing the deal on their official website.
Meanwhile, Donnarumma's move was being finalised in the early hours of Tuesday morning.
Confirmation of all three deals is expected in the coming hours.