Sporting have confirmed that their former winger will return to Portugal, while the Argentina international becomes the Red Devils' third major signing of the summer
United have been pursuing the Argentina international full-back for the majority of the summer and have now struck a deal to bring him to Old Trafford, which Goal understands is worth €20 million.
Nani's salary will be largely subsidised by United as they look to trim their squad ahead of their summer recruitment drive.
United have already signed Ander Herrera and Luke Shaw for a combined total of over €62m, and are now set to announce the signing of the 24-year-old.
Capable of playing as one of three centre-backs or as a wing-back in Van Gaal's favoured 3-5-2 formation, Rojo has emerged as a key target in recent weeks after Thomas Vermaelen's decision to leave Arsenal for Barcelona.
The star attempted to force through a move to United earlier this month by going on strike though he later issued an apology to his team-mates in an interview with the club's television channel.
"It was all a misunderstanding, I was a hothead. I'm sorry for what happened because I feel that I was unprofessional," he said.
"My career depends on Sporting. Now I will try to correct the situation. At the time Sporting signed me, it was not a good time. I can only thank Sporting.
"I want to tell the people that I didn’t behave correctly, but I’m am very fond of this club. They are the ones that will decide my carreer, and I hope it will be for the best both for me and the club. But I want to stress that I’m very happy at the club, and I like the way I’m treated and the care they give me."
Nani, meanwhile, has failed to impress Van Gaal during pre-season and will return to the club where he began his career before earning a €21m move to Old Trafford in 2007.
He has four years remaining on his current deal but made just 11 appearances under David Moyes and, later, Ryan Giggs in 2013-14.