Manchester City are among a number of clubs interested in signing Donnarumma, who at 18 is tipped to be the new Gianluigi Buffon of Italian football.
The goalkeeper is already AC Milan's No 1 stopper, having made his debut for the San Siro giants two years ago at the age of 16.
And he is currently deputising 39-year-old Buffon in the Italy national side.
Sky Sports Italia [football-italia] claim City have earmarked Donnarumma as their long-term option in goal, with the futures of Joe Hart, Willy Caballero and Claudio Bravo all in doubt.
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Manchester United manager Jose Mourinho is also eyeing a swoop for Donnarumma if his current No 1 David de Gea quits Old Trafford for Real Madrid this summer.
Mourinho and United worked closely with Donnarumma's agent Mino Raiola over Paul Pogba's world record transfer from Juventus last summer.
Raiola has gone on record saying there are offers for Donnarumma on the table – but that Milan have the chance to keep him.
Milan are hoping a takeover from Chinese businessmen will still go through despite the long-running saga hitting a number of hurdles in recent weeks.
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Manchester City reportedly want to sign Gianluigi Donnarumma this summer
Donnarumma (left) has been tipped to replace Buffon in the Italy national side
Milan want Donnarumma to sign a new contract
Donnarumma is supposedly waiting for the outcome of that takeover bid before deciding whether or not to sign a new contract in Milan.
This week the Italian spoke of his desire to remain at the club.
"I am tied to Milan," he said.
"I am Milan player and I hope to stay but as I said we will leave this with my agent."