The World Cup-winning Frenchman is set to complete a move to the Red Devils after the club agreed a £34m deal with Los Blancos, but quarantine and visa issues have thus far prevented him from linking up with the squad.
However, according to journalist Jonathan Shrager, Varane may be able to complete his medical and be unveiled by Man United on Wednesday, putting him in with an outside chance of featuring in their Premier League opener with Leeds United.
Shrager tweeted: "There is confidence that Varane's medical, signing and announcement can be officially completed on Wednesday, which would then allow him to train with the team on Thursday/Friday, opening up the possibility that he could feature in #MUFC's matchday squad in season opener vs Leeds."
Varane left Real Madrid after winning four Champions League crowns and three La Liga titles in a 10-year spell, as well as claiming 17 goals from 360 appearances.
The Red Devils have also bolstered their attacking line with the £73m signing of Jadon Sancho, who was pictured in training with United for the first time yesterday.