The France international remains a regular at the Bernabeu, where he has spent the last 10 seasons, but is due to become a free agent in a year's time.
Madrid must therefore weigh up whether to cash in on Varane this summer or potentially risk losing the centre-back for nothing next year.
Man United have been strongly linked with Varane over a number of transfer windows, without a move ever managing to materialise.
According to The Mirror, the Red Devils are still interested in the 28-year-old, but they again face missing out on his signature.
Chelsea are said to be willing to outbid their Premier League rivals with an offer that exceeds the £40m United are prepared to table.
The report indicates that the Blues feel a bid of around £60m will be enough to tempt Madrid to sell up, though they must also agree terms with the player. body check tags ::