Sporting have confirmed the sale of Marcos Rojo to Manchester United with Nani moving the opposite way on a season-long loan deal.
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.