Sporting said in a statement to Portugal's Market Securities Commission that the Premier League side would pay 20 million euros(16.03 million pounds)for the 24-year-old and the Portuguese side would not pay anything for a one-year loan for Nani.
If Rojo is sold on in the future for more than 23 million euros, Sporting will receive 20 percent of that fee, the club added.
Last week, Sporting ordered Rojo to train with the reserve team after club president Bruno de Carvalho complained he was trying to force the move and threatened to make him see out the remaining two years of his four-year contract.
Rojo had an excellent World Cup, part of an Argentina defence that conceded only four goals in seven games as they finished runners-up to Germany.
Raised at Estudiantes in La Plata, near Buenos Aires, he helped the club win their fourth South American Libertadores Cup and an Argentine league title in 2010.
He was drafted into the Argentina side in 2011 under Alejandro Sabella, who was also his coach at Estudiantes, and has been a regular since then.
"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," United said in a statement.
"A further announcement will be made when the process is complete."
Nani, 27, will return to the club he left seven years ago to join United.
He had a disappointing last season with United and also under-performed for Portugal at the World Cup, despite scoring against the United States.
(Writing by Brian Homewood; editing by Toby Davis)
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