The France international, whose contract at Bernabeu is due to expire in June 2022, is Ole Gunnar Solskjaer's top defensive target and sources have said that United are ready to move for the 28-year-old after receiving encouragement that Madrid are prepared to negotiate a realistic fee for the player.
United will complete their first major piece of transfer business this summer by signing Borussia Dortmund and England forward Jadon Sancho for £72.9 million ($100m) once the 21-year-old's involvement at Euro 2020 comes to a conclusion, but a move for an experienced defender is the next issue to be addressed by football director John Murtough.
When contacted by ESPN, a source at Manchester United confirmed there had been "some discussions" regarding Varane, but added that no time frame for a transfer is in place.
United had been considering Villarreal's defender Pau Torres as an alternative, but efforts to sign Varane have now been stepped up following France's surprise exit from Euro 2020 in the wake of Tuesday's round-of-16 defeat against Switzerland.
Despite Varane's diminishing contract, Madrid had initially valued the centre-half at €70m ($83m) -- a figure that United regarded as too high for a player entering the final year of his contract.
But with Madrid needing to offload players to fund a rebuild of Carlo Ancelotti's squad, United are confident that a deal can be done for a fee in the region of €50m ($59m).
Madrid could use the money raised by the departure of Varane to move for Torres, who has a €65m ($77m) release clause at Europa League winners Villarreal.
Madrid's financial position is complicated, and a "galactico" signing would mean Torres would have to wait until 2022.
Sources told ESPN the defender remains a Madrid target though, "whether it's for now or for next year," and Torres' preference would be to stay in Spain rather than try the Premier League.
Villarreal have also received interest in star forward Gerard Moreno, but the club are in a healthy financial state and do not need to offload players this summer.
ESPN's Alex Kirkland and Rodrigo Faez contributed to this report