Spanish newspaper Marca says Di Maria will cost United in the region of €70 million (£56m) and that the transfer “is very close to being closed”.
In England, Sky Sports report Di Maria has chosen to move to Old Trafford despite interest from Paris Saint-Germain in France and will take the fabled No. 7 shirt which has been worn by Cristiano Ronaldo, Eric Cantona and George Best.
The 26-year-old was linked with an exit from Madrid last summer following the £86m signing of Gareth Bale but it was Mesut Ozil who was sacrificed, with Di Maria going on to play a key role in Madrid's Champions League and Copa del Rey successes.
However, another summer of expensive recruitment - with James Rodriguez and Toni Kroos coming in - means Di Maria has been judged as expendable, despite claiming 17 assists in 34 Liga games last season.
It is believed the opportunity to work with Louis van Gaal was a key factor in his decision and Sky Sports says the deal will be completed when agent Jorge Mendes returns to Europe from Singapore.
Di Maria joined Real Madrid from Benfica in 2010 and was initially used as a winger. However, he was brilliant last season in a more central role under Carlo Ancelotti.
Angel Di Maria was the matchwinner for Argentina against Switzerland
After plenty of jokes at Ed Woodward's expense, it appears the under-fire United vice-chairman is finally coming up with the goods. Di Maria is a world-class star at one of the biggest clubs in the world and, with PSG also interested, getting the deal done is no mean feat. It's too early to say for sure, but it looks like United have landed a wonderful player.
WILL UNITED HAVE TO CHANGE FORMATIONS?
Di Maria was brilliant for Madrid last season but usually performed as one of a midfield three alongside Xabi Alonso and Juan Mata. In United's 3-5-2 he might be a bit exposed as one of a central two with Ander Herrera, and could play as a wing-back in theory, but the most likely outcome is that Van Gaal will switch to a 4-3-3 formation to best utilise his new player, should Di Maria join.
THE VIEW FROM SPAIN
Ivan Castello, Eurosport Spain: Although it seems strange, it happens sometimes with certain players. Di Maria is considered controversial because since the first season has been asking for a salary increase. Florentino Perez decides everything and believes that the Argentine's time at Madrid is over. Di Maria is a player who is in the prime of his career now. He was at his best in the last few months of last season in Madrid and at the World Cup with Argentina. Argentina probably would have been world champions without his injury. He is very fast on the left, he always plays with the group and he will give Manchester United a qualitative leap.
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