The France international has been tipped to quit the Bernabeu, with the Red Devils keen on the centre-half.
Jose Mourinho is said to be a big admirer of the 24-year-old, with the Special One handing Varane his debut at Real.
And according to the bookies, United and PSG are leading the race to sign the defender this summer, with Liverpool also enquiring about Varane.
BetStars spokesman Ian Marmion said: “Manchester United and PSG are going head-to-head in a battle for Real Madrid centre-back Raphael Varane.
“The two clubs are priced at 4/1 apiece to be the French defenders next club.
“United are keen to reinforce their back line and have struggled with injuries to their key defenders this season.
“Varane would be keen to reunite with Jose Mourinho – who handed the 24-year-old his first game at Real Madrid.
“The defender could also be tempted with a move back to his home country and PSG are first in line should Varane head to France.
“Manchester City and Liverpool have also expressed an interest in the Frenchman and are priced at 6/1 and 9/1 respectively.”=========================================================