Ole Gunnar Solskjaer's side revealed at the end of July that they had agreed a fee, which could rise to £42m, with Real Madrid for the transfer of the 28-year-old.
The deal has been held up due to the France international needing to quarantine before undergoing a medical, but Varane underwent the first part of his medical earlier this week.
According to Italian journalist Fabrizio Romano, the World Cup winner has now completed all the required tests and will sign a contract at Old Trafford until June 2025, with the option of a further 12 months.
Man United are hopeful that Varane, who has not been in action since Euro 2020, will be available for Saturday's Premier League opener against Leeds United at Old Trafford.
Jadon Sancho has been training with the 20-time English champions this week, meanwhile, and the England international looks set to make his first outing for the club against Marcelo Bielsa's side.
Edinson Cavani, though, is expected to miss the start of the 2021-22 campaign, having not yet returned to England following his involvement for Uruguay at the Copa America.