Varane's current deal at the Estadio Santiago Bernabeu runs until June 2022 and there are suspicions running through the club's offices that the Frenchman isn't planning on renewing it.
If, in the next few weeks, the World Cup winner reveals his intentions to run his contract down, Real Madrid will look to find a buyer for him in the summer to avoid losing him for free the following year, says Marca.
However, with a few months between now and the summer transfer window, the club's priority is still to convince him to stay.
Varane is being linked with Manchester United and signed with sports agency giants CAA Sports, who are very active in English football. His primary representative remains his brother Anthony, though.