Sporting, which is listed on the stock exchange in Lisbon, announced in a regulatory filing that United will pay 20 million euros ($27 million) for the 24-year-old Rojo. United confirmed the deal on Twitter but said it was subject to a medical examination being conducted and personal terms being agreed.
Rojo has played 28 times for Argentina, including the 1-0 loss to Germany in the World Cup final last month.
United's need to strengthen in defense was highlighted Saturday when the team opened the Premier League season with a 2-1 loss to Swansea in Louis van Gaal's debut as manager.
The 27-year-old Nani is heading home for the 2014-15 season, seven years after joining United, Sporting said.