The 29-year old has been given permission to leave the club on Wednesday and return to Argentina until the summer.
He has been convinced by Estudiantes chairman and former United midfielder Juan Sebastian Veron that he can revive his career there and claim his place back in the national side ahead of this summer’s Copa America.
Rojo joined United in 2014 from Sporting Lisbon, but he has struggled for game time under Ole Gunnar Solskjaer.
He saw a move to Everton fall through on deadline day in August and has watched predominantly from the sidelines as United have laboured through the season.
He was reluctant to return to Sporting as part of a deal for Bruno Fernandes, but is eager to return to Argentina where he will look to rebuild his reputation.