Goal understands the Red Devils had agreed a £20 million deal for the Argentine, with winger Nani going in the other direction on a season-long loan - and Rojo has seemingly confirmed the imminent move.
"It feels like a dream, to be a Manchester United player," the defender told ARG Radio Continental. "Leaving Sporting wasn't easy. I am very proud of having the chance of working with Van Gaal.
"[Juan Veron] told me about them when we played for Estudiantes."
Rojo previously made headlines after reportedly walking out of training when Sporting rejected United's initial bid, but apologised to his team-mates and supporters for his "unprofessional" conduct.
"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," he continued.
"If I made a mistake, I have already apologised because I didn't want to hurt the feelings of anyone at the club."
The 24-year-old, who reached the World Cup final with Argentina this summer before his side eventually lost to Germany, can play as one of three centre-backs in a 3-5-2 or on the left side of defence.
Rojo played for Sporting over 60 times during his two-year spell in Portugal, having previously turned out for Spartak Moscow and Estudiantes, with whom he won the Copa Libertadores in 2009.