The 25-year-old was a key member of the squad that won three consecutive Champions League titles under Zinedine Zidane but is understood to want a new challenge after a chaotic season in the Spanish capital.
Marca reported on Friday that the centre-back has grown tired of life in Madrid and told fellow Merengues players of his desire to leave, initially breaking the news to those closest to him before subsequently telling the squad at large.
But Varane, who has frequently been linked with Manchester United, refused to respond when questioned by reporters about a possible exit.
"I don't want to talk about my future," he told Le Parisien. "I'm not one to discuss my future publicly."
Zidane returned to the Madrid bench this month for a second spell in charge after first Julien Lopetegui and then Santiago Solari were sacked, with the club eliminated from Europe by Ajax and 12 points adrift of leaders Barcelona in La Liga.
The Frenchman is understood to want to keep working with his compatriot after getting the best from him during his previous spell in charge at the Bernabeu.
Varane has a release clause of €500m (around £430m) inserted into his contract and will therefore need the club to accept a lower bid if he is to seal a departure in the off-season.
