According to John Percy in the Telegraph, Leicester are hoping to persuade the Algerian international to stay "for at least another 12 months" after he helped them win the Premier League last season.
Percy added that Mahrez, 25, has previously admitted it is "possible" he could leave the King Power Stadium this summer, but if he does opt to depart, "Leicester will demand a figure of over £30 million."
That is more than affordable for a club of Real Madrid's stature, and Los Blancos were previously reported to have offered Mahrez a deal as they looked to make him "the latest Galactico signing at the Bernabeu," per France Football (via Metro's Mark Brus).
A Foxes exodus could be on the cards despite their Premier League title win last season, with Jamie Vardy reportedly concerned the likes of Mahrez and N'Golo Kante could follow him out of the club if he opts to join Arsenal, per Bleacher Report's Dean Jones:
Certainly, Mahrez could not be blamed if he does have his head turned by Real, as the Spanish giants can offer him more guaranteed long-term success than Leicester.
Real's position as perennial contenders for both La Liga and the Champions League could be the key to persuading Mahrez to leave Leicester.
He only emerged last season as a true star—the Foxes signed him for just £400,000 from Le Havre in January 2014—and could slip out of the limelight if Leicester fail to repeat their heroics of 2015-16 in the coming season.
Mahrez is more likely to win multiple trophies if he moves to Real, and he could fit in well in manager Zinedine Zidane's system as he has both terrific flair and an excellent work ethic.
His phenomenal return of 17 goals and 11 assists in the Premier League last season saw him deservedly crowned PFA Player of the Year, and he could likely be even more prolific at Real playing alongside the likes of Gareth Bale and Cristiano Ronaldo, per WhoScored.com.
However, Mahrez could slip away from Real's grasp if he decides to take Leicester up on their new contract offer.
Meanwhile, Real are reportedly set to send out 23-year-old midfielder Lucas Silva on loan again after he spent last season at Marseille.
According to Marca, "up to nine" clubs from Germany, Portugal, Italy and Spain are interested in taking the Brazilian on a short-term deal with Real unwilling to sell Silva.
The report also added that Zidane is "against the idea of including a buyout clause in Lucas' contract," as he believes that he can improve enough to earn a long-term spot in the Real side.