United has been pursuing the Argentina international fullback for the majority of the summer and has now struck a deal to bring him to Old Trafford, which Goal UK understands is worth 16 million pounds.
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Capable of playing as one of three center backs or as a wing back in Van Gaal's favored 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 holding out, though he later issued an apology to his teammates 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 preseason and will return to the club where he began his career before earning a 17 million pound 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.