Goal reported on Monday that the Red Devils had agreed a €20 million fee for the star with the Lisbon side, with winger Nani going in the other direction on a season-long loan, and according to Rojo, the deal is now done.
"It feels like a dream, to be a Manchester United player," the defender told ARG Radio Continental. "Leaving Sporting wasn't easy."
Rojo previously made headlines by going on strike before apologising to his team-mates and supporters for his 'unprofessional' conduct.
The 24-year-old, who reached the World Cup final with Argentina this summer before his side eventually fell 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. He was named in the Fifa World Cup All-Star side for his performances in Brazil this summer and will shore up a Manchester United defence beset by selection problems of late.