The Daily Mail and The Sun both report that the winger punched reserve-team player Davide Petrucci in an astonishing incident this week.
The pair exchanged words after a hefty challenge in training at Carrington.
Nani is said to have confronted Petrucci in the gym later and appeared to land a punch on the 20-year-old Italian.
It is the latest episode in a sorry year for Nani, who has lost his status as a first-team regular for United.
The Portuguese international stayed at Old Trafford in the summer despite speculation linking him with a move away - a £25m move to Zenit St Petersburg reportedly stalled over wages.
He has failed to turn his fortunes around, and this week's incident could persuade manager Sir Alex Ferguson that the 25-year-old's future lies elsewhere.
Ferguson has not yet decided what action to take, but a parting of ways seems ever-more likely.
Nani's contract expires at the end of next season, and talks over an extended deal have broken down. United would look to sell rather than face the prospect of losing a highly-rated asset on a 'Bosman' free transfer.
The club have not commented on the training-ground incident.
Meanwhile, Ferguson Alex Ferguson has admitted he never felt Nemanja Vidic had recovered full fitness after his cruciate ligament injury.
The United skipper attempted his comeback in pre-season after snapping his ligaments in Basle last December. But just four games into the Premier League campaign, the Serbian has been forced to go under the knife again for further surgery that will keep him out for two more months.
It is a grievous blow to United, who are already without Chris Smalling and Phil Jones and are left with just two fit senior central defenders, but Ferguson is not entirely surprised at the turn of events and he said: "I don't think he was ever 100 per cent. I noticed a couple of times in training that he was limping a little bit. Then he started feeling tightness in his knee."
He added: "He wanted to carry on and in that situation that is what tough guys can do.
"Obviously the main operation was to correct the cruciate but when you do cruciates there is always the danger that the cartilage or meniscus is damaged also and that was the case."
It means Rio Ferdinand will partner Jonny Evans against Tottenham at Old Trafford this evening, with Birkenhead-born rookie Scott Wootton acting as cover.
Under the circumstances, it is understandable why Ferguson might hope Roy Hodgson does not call Ferdinand back into the England fold when he unveils his squad for the World Cup qualifiers with San Marino and Poland next Thursday.
Ferguson said: "I wouldn't think he'll get called up. I think Roy made his decision for the Euros and I can't see him changing that.
"It would be difficult for him to go to Rio now and welcome him back. If that happens, it's entirely up to Rio in terms of what he wants to do."
At least United have an unbeaten home record against Tottenham that dates back to 1989 to comfort them. And there is also the potential for Wayne Rooney starting alongside Robin van Persie for the first time, even if Ferguson insists he does not know what his best strike partnership is going to be.
"It is early doors but at the moment I don't know where I am going in the sense of what my best partnership will be," said Ferguson. "Probably the strongest part of the club will be in the striker department. It (Van Persie's arrival) should give everyone a push."
"Danny Welbeck is in the same position as Javier in that they are not the first-team choices at the moment. That doesn't mean to say it will stay that way because competition is always healthy and them being the kind of young lads they are will be desperate to establish themselves."