United announced on July 27 that a deal for Varane had been agreed but a medical and personal terms had yet to be finalised. Sources told ESPN a deal worth approximately €41 million had been agreed.
United unveiled Varane, who penned a contract to 2025, ahead of their game against Leeds United on Saturday. He wasn't in the squad for the match, which his new team won 5-1, as he hasn't trained with the side yet.
"There is a lot more that I want to achieve in my career and I know I'm joining a squad full of great players who will all have the same determination to win matches and trophies. Having spoken to the manager I can see how much progress has been made over the last few seasons and I now join a group ready to play at the highest level.
"I want to make an impact here and I will give everything to become part of this club's illustrious history."
Varane was in the final year of his contract at Madrid and the club opted to sell him this summer rather than lose him on a free transfer later down the line.
Sources told ESPN that despite this success, he was looking for a new challenge in the Premier League.