The Dutchman, who has long been linked with a move away from the Bernabeu this summer, is believed to have agreed a five-year contract that will earn him €4 million per season.
The Nerazzurri are in need of a new midfield recruit following the loss of Argentine international Esteban Cambiasso through injury, with Sneijder's Real Madrid team-mate Rafael van der Vaart also tipped to join Jose Mourinho's side.
Sneijder has made 52 league appearances - netting 11 times - for Real since joining from Ajax in 2007.
But following the expensive acquisitions of Kaka, Cristiano Ronaldo, Karim Benzema, Alvaro Arbeloa and Xabi Alonso, Real are keen to recoup some cash, with Sneijder, Van der Vaart, Arjen Robben, Roysten Drenthe and Ruud van Nistelrooy all available for transfer, while Klaas-Jan Huntelaar has already joined AC Milan.
The former Ajax starlet - who has been capped 53 times by his country - has been linked with Chelsea, Arsenal, Tottenham Hotspur, Manchester United and Aston Villa during the close season.
He had seemingly been stripped of his No.10 shirt, after Lassana Diarra was spotted wearing the same number in Monday's friendly against Rosenborg.
And despite the player's repeated pleas to stay with the Madridistas and fight for his place in 2009/10, he now appears to have conceded defeat, and will be plying his trade in Serie A.
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