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'.