Ligue 1 club have now admitted dialogue has been opened for the defender, who yesterday, told Goal.com of his passion for the Spanish outfit
Lens president Gervais Martel believes a deal could be completed soon, leaving Manchester United, and Paris Saint-Germain empty handed.
Speaking to the club's official website, Martel said talks were gaining momentum.
"It's not done officially with Real Madrid but talks are on track and hopefully it [the transfer] will happen today," Martel said.
On Tuesday, Varane spoke of his admiration for Real Madrid in an exclusive interview with Goal.com. Reports followed suggesting Zinedine Zidane's influence would help push a deal through.
Martel believes Zidane has won Varane over and he appeared grateful to Madrid.
"Madrid directors were very good to us, they have been speaking with the player. There is nothing that could go wrong," he added.
Critics have tipped the defender to struggle at the Bernabeu, but the president believes he will thrive.
"Some believe that Raphael will not play there. I can tell you he will not go to Real Madrid to repaint the changing rooms! Raphael is intelligent, everybody knows it. I want to emphasise the speaker was exceptional, in this case Zinedine Zidane."
Previous reports suggested United had made a €10.5 million bid for the player, but Martel did not mention the Premier League side as Varane edges closer to Spain.