The 28-year-old could fly to Manchester on Saturday but Varane has to obtain a visa before he can fly to England, says the Manchester Evening News.
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.
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!