The 26-year-old Argentina international is expected to fly into Manchester on Monday to hold talks with United next week with the hope of being ready to face Burnley at Turf moor next Saturday.
When the transfer is finalised, the fee will be a record in Britain, eclipsing the £50m Chelsea paid for Fernando Torres three years ago.
According to Sky Sports’ Spanish football expert Guillem Balague, United initially offered €60m and Madrid wanted €90m, with the two clubs eventually agreeing on €80m (£63.9m).
Manchester United have declined to comment on whether a fee has been agreed.
After watching United stumble to a 1-1 draw at Sunderland on Sunday afternoon, Gary Neville told Sky Sports: “It’s good business. United have lost their negotiation strength but they just need to get the right players. This one is.
"He'll get pace into the game and get from box to box. He’s the right signing. You could argue they need other players in other areas but they definitely need a wide player.”
United manager Louis van Gaal refused to confirm or deny a deal had been agreed for the Argentine in his post-match interview with Sky Sports.
“The club has to announce it,” he said. “When we are thinking it is the right time for a transfer, then we come to you and say ‘OK, we have that player or the other player’. But at this time we cannot say anything.”
When asked if Di Maria is a player he would like to sign, Van Gaal confirmed he is but added: “I like Messi also and I like Vidal also so we have a lot of players in this world that I like but they cannot come always to Manchester United.”
Madrid manager Carlo Ancelotti confirmed on Sunday the player had come to the club’s training base to bid farewell to his team-mates and the club’s staff.
"Di Maria has not trained with us and he came in to say goodbye to the players and people at the club," Ancelotti said.
"We are thankful for what he has done at this club. There is nothing official yet but it is being sorted out.”
Despite winning man of the match in Real’s Champions League final triumph over city rivals Atletico last season, Di Maria was deemed expendable by club president Florentino Perez following the arrival of World Cup top-scorer James Rodriguez from Monaco.
That put Di Maria’s place in Ancelotti’s starting 11 in jeopardy and, when contract talks between the player and club stalled, with Di Maria feeling club officials were failing to recognise his value and contribution to the team, the decision was made to sell.
The Real manager maintains the decision to leave was made by the player and hinted the Spanish giants will be looking for a replacement before the window closes.
"The decision is his and the club has done what it could to keep him here but he thought differently and so good luck to him,” Ancelotti said.
“I am very happy with the work that he has done for Madrid but we have the chance to replace him well.”