The Red Devils’ total outlay on the France defender could reach £42m.
Varane is United’s third summer signing following the arrivals of Jadon Sancho from Borussia Dortmund and goalkeeper Tom Heaton.
Varane joined Real Madrid from Lens in the summer of 2011, and the 28-year-old enjoyed enormous success with Los Blancos, winning four UEFA Champions Leagues, three La Liga titles and a host of other trophies. He also won the 2018 FIFA World Cup with France in Russia.
Varane said: "Manchester United is one of the most iconic clubs in world football and the chance to come here and play in the Premier League is something I couldn't turn down.
"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."
United manager Ole Gunnar Solskjaer said: "I'm absolutely delighted that we have managed to secure one of the best defenders in the world over the last 10 years. Raphael is a proven winner who we have tracked over a long period of time and we know just how much of a dedicated professional he is.
"We have great depth of international defenders and he will add his immense skill set and leadership to that group. He is a unique defender with a rare combination of top-level attributes that I know will rub off on our younger players.
"He has won everything there is to win but I know that he is still determined to succeed, I can't wait to welcome him into the squad."
