Los Blancos are looking to bolster their attack in the ongoing transfer window and have identified the former Leicester City star as a potential candidate.
Real Madrid have touched base with Mahrez’s entourage to gather information, according to French outlet Foot Mercato.
However, Zinedine Zidane’s side could only make a formal offer once they have resolved Gareth Bale’s situation at the club.
The Spanish giants have been looking to move the Wales international on, but the forward is happy to see out his contract at the Bernabeu.
If Bale leaves, Real Madrid may move for Mahrez.
However, it is unclear whether Manchester City would be willing to let the winger leave and he has a contract at the Etihad Stadium which still has a further three years to run.
Real Madrid are reluctant to make any expensive signings over the course of the transfer window and will wait until next summer to splash the cash.