Sporting Lisbon on Tuesday night sanctioned the sale of the Argentina international to United, with Portuguese winger Nani returning to his former club as part of the deal.
Centre-back Rojo, 24, will become a United player should he pass a medical and agree personal terms, the Barclays Premier League club announced through their official Twitter feed.
Sporting welcomed Nani back to the Estadio Jose Alvalade and revealed that they will use nine million euros (£7.2m) from the reported £16m fee for Rojo towards the construction of their new indoor arena, the Pavilhao.
Rojo is equally comfortable at centre-back as he is at left-back and played all 120 minutes of July's World Cup final as Argentina lost to Germany after extra-time in Rio de Janeiro.
In an interview with the radio station Continental in his homeland, quoted in several national newspapers, he said: "It's a dream to play at Manchester United and I am very proud of having the chance of working with Van Gaal.
"I spoke with Juan Sebastian Veron about Manchester United when we were at Estudiantes
I have always liked English football, and I should adapt to this new playing style easily."
United's move for Rojo had been complicated by a row between Sporting and a third-party ownership group, while the player upset his employers by initially refusing to train in a bid to push through the deal, although he later apologised and returned to work.
"There was a misunderstanding, I never wanted to do something wrong at Sporting as I have always been very thankful to that team that gave me many chances for playing in Europe," Rojo said.
"If I made a mistake, I have already apologised because I didn't want to hurt the feelings of anyone at the club."
The 27-year-old Nani, who moved to Manchester seven years ago, is understood to have joined the Lions on a season-long loan, but neither club has released details pertaining to the winger's move.
The Cape Verde-born player spent four years as a Sporting player after leaving Real Massama in 2003.
He has made 228 appearances in all competitions for United, scoring 40 goals.
Source : PA
Source: PA