The Argentina international has been targeted by Southampton this summer but has learned of the Old Trafford club's interest and is keen to link up with Louis van Gaal.
Southampton has targeted Rojo for much of the summer and has been working hard to broker a deal with Sporting Lisbon while also attempting to persuade the player to move to St Mary's Stadium.
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The south coast club was prepared to offer 20 million pounds for the Argentina international but Sporting stood firm on his £30 million release clause.
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United has now entered the fray and made an offer of around £20 million and, while it is still some way short of what the Portuguese giant wants, Rojo has learned of the Old Trafford club's interest and is very much keen on the move.
Rojo, who can play centrally or on the left, impressed during the World Cup and was a regular feature for Argentina in its run to the final, where the team lost to Germany.
A source confirmed to Goal: "Rojo knows United have now come in and things have changed - he wants to move to Manchester."
United has brought in Ander Herrera and fullback Luke Shaw this summer as Louis van Gaal prepares for his first season in the Premier League. Van Gaal has told club officials that he is happy with his squad but the Red Devils could bring in more signings before the transfer window closes.
United missed out on the signing of center back Thomas Vermaelen, who opted to leave Arsenal for Barcelona, but Goal understands that chief executive Ed Woodward has held talks with Nigel de Jong's representatives over a move for the AC Milan midfielder.