The Gunners have been heavily linked with the France international over the course of the past few years, with Arsene Wenger believed to be a huge fan of his compatriot.
A big money offer was rumoured to have been made last summer, but Los Blancos swiftly rebuffed the approach as they elected to stick by Benzema after a difficult period for the player.
But this transfer window could be different, with the Spaniards thought to be ready to cut their forward loose as they step up their own plans for additions.
Real are keen on both Liverpool’s Luis Suarez and Sergio Aguero of Manchester City, yet they know they must sell before they can buy either.
Benzema has been jeered by the club’s fans of late following his failure to live up to the potential he showed with Lyon prior to his switch to the Bernabeu, which has further fuelled talk that he could be on his way in the coming months.
Wenger is said to have a big budget at his disposal to bolster his squad, and may look to swoop for the £25m-rated hit-man to address a need for a s top class goal-getter.