Spanish football expert Guillem Balague said the Argentina winger has been sold on a switch to Old Trafford by Cristiano Ronaldo.
The journalist said talks are at such an advanced stage it has been agreed Di Maria will wear the No 7 shirt once sported by the Portuguese and a string of illustrious predecessors.
The fee, according to Balague, will be between €60-70 million (£48-55m).
Balague told Sky Sports: "The final meetings will take place at the beginning of next week.
"So advanced are the negotiations that it has already been agreed that Di Maria will wear the No 7 shirt, worn in the past by no lesser players than David Beckham, Eric Cantona, Ronaldo, George Best and Bryan Robson."
Balague said Di Maria's agent Jorge Mendes was on his way from Singapore to Europe with a view to sealing Di Maria's departure from the Bernabeu.
The journalist added: "As soon as Di Maria knew of Manchester United's interest he was very excited about the possibility of coming to Old Trafford.
'At first Paris Saint-Germain showed interest in singing him and said they were prepared to meet his wage demands of €8m a year net but the deal offered by United was much more interesting to him.
"Firstly, United is a club with much more prestige throughout Europe, and secondly, it's a new project where he knows he will be one of the leaders.
"Cristiano Ronaldo has also told him some great things about United and has succeeded in persuading him that it is a good option even though the club are not playing in Europe this season.
"He is also very excited about the prospect of working with Louis van Gaal and everyone is expecting that the deal will be closed soon."