Raphael Varane hopes to travel to Manchester this weekend to complete his transfer to United.
The 28-year-old could fly to Manchester on Saturday but Varane has to obtain a visa before he can fly to England.
Varane will have to quarantine following his arrival and will then undergo a medical to complete his £43million move.
The France international bade farewell to Real Madrid players and staff on Friday at the club's Valdebebas training ground.
Varane could start training with his new teammates next week ahead of a possible first appearance for United in their final pre-season friendly with Everton at Old Trafford on August 7.
"Of course, very happy that we've managed to get a deal with Real Madrid," United manager Ole Gunnar Solskjaer said. "Medical should hopefully be okay, he's a proven winner, he's a player we've followed for many, many years."
Solskjaer believes Varane and £72.9m arrival Jadon Sancho will make a 'huge difference'.
"Sir Alex was very close to signing him. This time around, we've got him at the other end of his career. He's proved over his old career, what a good person he is, professional and he's won what there is to win.
"I'm very happy getting both our players in. The club shows ambition with one of the most exciting young forwards in world football and one of the most respected centre-backs in the world and one of the most winning centre-backs in the world.
"With him in, we've got different ways of playing as well. I can see different ways of playing with Raphael, I can't wait to get him in, hopefully we can get him in as soon as.
"From what I know, the club is working on the quickest possible solution to get everything sorted [for the medical]. I don't know the timescale yet, there's a visa coming into England now with Brexit and quarantining, everything has to be done as quickly as possible and by the book.
"As I've said before, every manager would be happy the sooner and the earlier the deals are done. This year we've managed to get it sorted, the two we've got in will make a huge difference for us, both this season, short-term, but also long-term."
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