The Daily Mail says the France international is the subject of long-held interest from United and they have been given encouragement he is ready to leave Madrid at the end of this season though there could be competition from Paris Saint-Germain, the club Varane followed as a boy.
The 27-year-old Varane has one year left on contract after this summer and is still valued in the region of £60million by Madrid who will need to raise funds for their own rebuilding.
United would hope to get that figure down but there is a growing recognition within the club that Varane fits the profile of defender they need to play alongside Harry Maguire.
Of the alternatives, Villarreal's Pau Torres is talented but still raw while Sevilla's Jules Kounde is regarded as a defender of huge potential but there are reservations over whether he has the best physical attributes to compete in the Premier League, plus a buy-out clause of £70m. Fiorentina's Nikola Milenkovic has also been offered.