Donnarumma, 18, has been heavily linked with a move to Old Trafford in recent weeks with David De Gea looking destined to leave for Real Madrid.
Jose Mourinho has outlined his intentions to keep hold of De Gea beyond this season.
But the Spain international has failed to feature for United in their last three games, including the Red Devils' Europa League triumph against Ajax yesterday.
And last week speculation over his departure intensified after he posted a picture of him applauding fans.
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Manchester United are ready to make a £65m bid to sign AC Milan keeper Gianluigi Donnarumma.
Alongside the image he wrote: “Thank you for your unconditional support.”
With Real set to go all out to sign De Gea, Mourinho has had to cast his eye on potential replacements.
And Italian outlet Telelombardia claim United are ready to make a big-money move for Gianluigi Donnarumma should De Gea sign for the La Liga giants.
Donnarumma has already firmly established himself as the No 1 shot-stopper at Milan and his impressive displays have apparently attracted interest from Mourinho.
Manchester United are ready to make a £65m bid to sign AC Milan keeper Gianluigi Donnarumma
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The Italy international's contract only runs until 2018 and his agent Mino Raiola, who also represents Zlatan Ibrahimovic and Paul Pogba, reportedly wants to get the best deal for him.
Telelombardia say Milan are determined to keep hold of the youngster and will look to offer him a contract extension.
But with Raiola having a good rapport with Manchester United through the transfers of Ibrahimovic and Pogba, he could be tempted to broker a deal with the Red Devils.
Donnarumma has made 37 Serie A appearances for Milan this season and has kept 12 clean sheets.