This is according to Spanish newspaper AS, who understand the Red Devils have made enquiries for Cristiano Ronaldo, Gareth Bale, Karim Benzema and, most recently, Raphael Varane.
Varane did not enjoy the best end to the season with Real Madrid - the France international suffered an untimely dip in form - and United saw this as the perfect opportunity to test the waters.
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As it stands, Real Madrid have little interest in selling the 22-year-old, who is widely considered the best young centre-back in world football.
In order to even tempt Los Blancos into selling, Manchester United would need to produce a huge fee. The Spanish giants are under no pressure to offload the Frenchman, who signed a new six-year contract last September.
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Small chance Benzema will leave
Benzema is contracted to the Bernabeu until 2019 and will continue at Real Madrid unless newly-appointed manager Rafael Benitez decides he does not want the French striker.
It’s no secret that Real Madrid president Florentino Perez is a huge fan of the 27-year-old, who is likely to remain the club’s first-choice centre-forward next season unless Benitez can convince Ronaldo to switch positions.
It was reported last week that Gareth Bale is unhappy playing on the right wing and has told Real Madrid bosses that, instead, he wants to play on the left - Ronaldo’s position.
The Wales international feels his effectiveness is limited when deployed on the right flank and has left it down to Benitez to find him a new position in time for the start of the 2015-16 campaign.
Ronaldo and Bale to stay with Madrid
Manchester United are interested in both Bale and Ronaldo but have very little chance of signing either superstar this summer.
Ronaldo, for certain, is going nowhere this summer. The three-time Ballon d’Or winner has another three years left on his contract at the Bernabeu and plans to stay with Real Madrid for the foreseeable future.
Bale’s situation is a little more complicated following such a frustrating second campaign at Real Madrid, but Perez is desperate for Benitez to get the best out of the world’s most expensive footballer next season.
Furthermore, Bale has no desire to move elsewhere and is keen to prove his worth after losing the faith of many Real Madrid supporters.
Manchester United fans: which of the four Real Madrid stars does your team need most? Tell us by leaving a comment in the box below!