The Red Devils have been strongly linked with the France international over the last few weeks and are now in the final stages of securing his signature.
Varane needs to finalise personal terms with the 20-time English champions before undergoing a medical, and it is believed that official confirmation of the 28-year-old's arrival will arrive next week.
Man United will want the Frenchman to be in the XI for their 2021-22 Premier League opener against Leeds United at Old Trafford on August 14.
However, the pace of the deal has been complicated due to the fact that the former Lens youngster will need to quarantine upon his arrival in the United Kingdom due to coronavirus restrictions.
Various reports have suggested that Varane will arrive in England later this week, although The Athletic claim that a particular date for the defender's medical has not yet been set due to the quarantine restrictions.
Varane only recently rejoined his Real Madrid teammates, having been given time to recover following his participation for France at this summer's European Championship.
Man United, who are believed to be paying £34m plus add-ons for the centre-back, will continue their preparations for the 2021-22 campaign against Brentford on Wednesday night before taking on Preston North End on Saturday.
Varane, of course, will not be involved in either of the English club's next two matches, but there is a chance that he could feature in the friendly against Everton at Old Trafford on August 7, providing that the details surrounding the transfer are able to be moved on over the next couple of days.
An appearance against Everton on August 7 would then place the defender in a strong position to make his Premier League debut against Leeds at Old Trafford on August 14.
Varane, at that point, would have had very little time with his new teammates, but he did make four appearances for France at Euro 2020 and should therefore not have lost too much fitness.
Red Devils boss Ole Gunnar Solskjaer will be hopeful of selecting Harry Maguire alongside Varane as part of a new-look centre-back partnership at the start of the new season, while recent arrival Jadon Sancho is also pushing to be involved in the starting XI on the opening weekend of the 2021-22 campaign.
Potential first-team debut: August 7 vs. Everton (friendly)
Potential competitive debut: August 14 vs. Leeds United (Premier League)