Argentina international Rojo, 24, had hinted last week that he was poised to move to England and become part of Louis van Gaal's squad before Sporting rubber-stamped his departure on Tuesday.
A United statement read: "Manchester United is delighted to announce it has reached an agreement with Sporting Lisbon to sign defender Marcos Rojo, subject to a medical and agreeing personal terms.
"A further announcement will be made when the process is complete."
Rojo had returned to training with the Portuguese giants only after apologising to them for previously refusing to play, as he looked to force through a transfer.
In a recent interview, Rojo had said: "It's a dream to play at Manchester United and I am very proud of having the chance of working with [Louis] van Gaal.
Rojo, who can play at left-back or centre-back, claimed he always wanted to play for United, after speaking with former United star Juan Sebastian Veron earlier in his career.
"I spoke with Juan Sebastian Veron about Manchester United when we were at Estudiantes," he said. "I have always liked English football, and I should adapt to this new playing style easily."
Nani rejoins the Portuguese club that he left for Old Trafford in 2007 having scored 25 goals in 147 appearances for United.
United, who lost their opening league game against Swansea on Saturday, have been frustrated in their quest to strengthen a squad that has seen no new recruits since Luke Shaw and Ander Herrera arrived in June.