According to the Mirror, Nani, 25, had a training-ground bust-up with reserve side midfielder Davide Petrucci, which angered boss Sir Alex Ferguson. However, the Portuguese were not dropped for the 3-2 loss against Tottenham Hotspur at the weekend.
Meanwhile, the Turin club are ready to make an offer worth £12 million for the 25-year-old, half of United's £25 million valuation.
Ferguson was left furious after Nani missed the penalty against Galatasaray and is now increasingly frustrated after the winger punched 20-year-old Petrucci.
Nani's contract runs out in less than two years and he believes that the club has made false promises to offer him a new contract worth £120,000 weekly.
And now, United fear that Nani could run down the remainder of his contract and leave the Old Trafford club for free.
Russian giants Zenit St Petersburg made a £25 million bid earlier this month but Nani forced himself out of the move. The Portuguese has been out of form lately after a successful campaign last season.