He is eager to fight for his place in Manchester, but Spanish publication SPORT per Daily Express have said a move was mooted at one point.
The Blaugrana ‘studied’ sending defender Samuel Umtiti in the opposite direction as a way to balance their books, but United weren’t interested.
France international Umtiti has been plagued by injury in recent years and played just 13 times in La Liga last season.
He was once regarded as a top prospect but is now 27-years-old with a poor fitness record.
Among their summer arrivals so far are Sergio Aguero, Eric Garcia and Memphis Depay.
Barcelona finished third in La Liga last season and Ronald Koeman is keen to bridge the gap between themselves and Atletico Madrid.
Meanwhile, United have signed Jadon Sancho this summer and look likely to add to their defense.
The 28-year-old is entering the final year of his deal and is ready for a new challenge after a decade in Spain.
Madrid have already seen Sergio Ramos depart for Paris Saint-Germain this summer and want a fee for Varane.
United regularly paired captain Harry Maguire with Victor Lindelof last season but are aware they need more experience.
Varane has won four Champions League titles with Madrid as well as the World Cup with France in 2018.
Manager Ole Gunnar Solskjaer was asked about transfers following Sunday’s 2-1 win over Derby County.
:speaking_head: “We hope we can announce one player that’s been very much written about.”
Ole Gunnar Solskjaer says Jadon Sancho’s transfer hasn’t been a secret but refuses to answer questions about Varane pic.twitter.com/JW83RG0cLH
– Football Daily (@footballdaily) July 19, 2021
Sancho is set to be confirmed as United’s second signing of the summer after Tom Heaton, but Solskjaer kept his cards close to his chest.
The Norwegian coach told Sky Sports: “There are a couple of things to do with paperwork but we hope we can announce one player who has had a lot written about him. We are working, we want to have a strong squad.
“I don’t think it’s a secret we have a deal with Dortmund, but until the paperwork is done he is still a Dortmund player.”
When asked about Varane, he replied: “Same again.”