Southampton have targeted Rojo for much of the summer and have been working hard to broker a deal with Sporting Lisbon while also attempting to persuade the player to move to St Mary's Stadium.
The south coast club were prepared to offer €20 million [£16m] for the Argentina international but Sporting stood firm on his €30m release clause.
United have now entered the fray and made an offer of around €20m [£16m] and, while it is still some way short of what the Portuguese giants want, 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 their run to the final, where they 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 have brought in Ander Herrera and full-back 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 yet make further signings before the transfer window closes.
United missed out on the signing of centre-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.