The Frenchman has been an ever-present figure for Los Blancos in recent years, helping his side win three consecutive Champions League crowns.
But reports suggest Varane is ready to quit Madrid in search of a new challenge, despite the return of fellow countryman Zinedine Zidane this month.
United, PSG and Bayern Munich are all said to be in the hunt for the 25-year-old, with outlets in Spain suggesting the Red Devils are ready to fork out €100m (£86m) to acquire his services.
But when pressed on his Madrid future this week, Varane was not willing to divulge his plans for the coming months.
“I do not want to discuss my future,” he told Le Parisien.
“I do not usually express myself publicly about my future.”
As mentioned, Zidane has returned to the Santiago Bernabeu after Julen Lopetegui and Santiago Solari failed to achieve success at the Spanish outfit.
And Varane is happy to welcome his former boss back to Madrid.
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“Nine months is not a long time. It feels like yesterday that he left,” Varane added.
“I like him a lot as a manager and a tactician. He has been very important for me.
“It is funny, because there existed the same doubts about him before he started as a manager in January 2016, as there existed for me before the 2018 World Cup.
“He succeeded in putting the critics to one side, in a strong way and to show his aptitude to manager the biggest club in the world.”
United have already been beaten to the signing of one defender by Madrid in recent weeks.
Porto’s Eder Militao will join the La Liga giants in the summer after securing a €50m (£43m) deal.
United were rumoured to be interested in the 21-year-old while Jose Mourinho was in charge last year.