Both of the clubs see the French international as the ideal centre-half purchase, as they seek to prioritise that area in the summer.
For United, he is the top target behind Borussia Dortmund's Jadon Sancho.
The 25-year-old had been a key member of the Madrid team in an era that has seen them win four Champions Leagues in five years, and previously viewed as unsellable by the Bernabeu hierarchy.
With president Florentino Perez set to sanction the most drastic overhaul of the team since 2009, however, the club need to raise funds and sources say there is a feeling that "nothing can be written off" when the window opens.
Madrid are hoping to bring in Eden Hazard and one of Neymar or Kylian Mbappe, as well as at least one other high-profile signing, so are open to any big offers.
A willingness to let Varane go to PSG could well smooth the move of one of their two stars to the Bernabeu. It is also seen as telling that Madrid have not rejected the enquiries out of hand.
If it went to a straight purchase, Varane would cost in excess of £100m.
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