United have made no official comment on the matter, but it is understood a fee has been agreed between the Red Devils and Real that will eclipse the British record £50million Chelsea paid to Liverpool for Fernando Torres in 2011, with reports putting the figure at a minimum of £59.7m.
It is also understood Di Maria - who bid farewell to his Real team-mates on Sunday - was flying to Manchester on Monday ahead of a medical.
And the expectation is that the deal will be complete within around 24 hours, although not in time for the Argentina international to be involved in United's Capital One Cup tie at MK Dons on Tuesday night.
The 26-year-old would be set for a debut in Saturday's Premier League match at Burnley.
United boss Louis van Gaal would neither confirm nor deny anything with regard to the Di Maria situation after his side's 1-1 league draw at Sunderland on Sunday.
Van Gaal said: ''I cannot say anything about transfers, you have to understand that. When a transfer has been finished, then we can say something. But now you have to wait and see.
''When the moment is there, Manchester United shall announce it.''
He added with a smile: ''It is not only Di Maria - maybe (Arturo) Vidal or (Lionel) Messi! We are a big club.''
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- Fernando Torres
- Di Maria
- Manchester United