The two Premier League clubs are both interested in signing the 21-year-old France international, who is widely considered the best young centre-back in world football.
Real Madrid reportedly value the defender at £20 million - although that seems an awfully low price for a player with such talent and potential.
Varane: I'm happy at Real Madrid
Furthermore Varane signed a new six-year deal with the reigning European champions last September - and the player insists he is content at the Bernabeu, despite the constant talk over his future.
"I’m 100 percent with the team and feeling happy. I have a contract until 2020,” Varane told reporters. “I’m calm and when I’ve played, I’ve tried to do well. Sergio Ramos and Pepe play more than me, but I try to do the best I can.
“My knee hasn’t hurt for a year now. I can play every three days without discomfort. My most important objectives are La Liga and the Champions League."
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Varane: Real Madrid first-team regular
Varane played the entire 90 minutes during Sunday's 3-0 away win over Getafe. Despite the comprehensive scoreline, it took Carlo Ancelotti's side 63 minutes to break the deadlock.
But the Frenchman insists the La Liga leaders were not concerned following their frustrating first half at the Coliseum Alfonso Pérez.
“It’s good to have time to prepare for games," Varane added. "We weren’t worried at 0-0, we wanted to win and had the determination to do so."
Real Madrid unlikely to sell Varane
Real Madrid agreeing to sell Varane seems a far-fetched possibility at this moment in time - particularly for a fee in the region of £20 million.
Given his talent and the fact he is contracted to Real Madrid for another five years, the Spanish giants would surely dismiss any offers below £40 million.
Manchester United, who were keen to sign Varane before he joined Real Madrid from Lens in 2011, are huge fans of the defender and believe he could slot straight into the team's back-line.
Chelsea manager Jose Mourinho, meanwhile, handed the player his La Liga debut while managing Real Madrid - and may regard the player as an ideal long-term replacement for captain John Terry.
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