Earlier this season, Benzema penned a new long-term deal with Real, but speculation persists over his future at the Bernabeu.
The 30-year-old has scored just four goals in 15 outings in La Liga and the Champions League this season and according to Diario Gol, Arsenal may look to tempt Real into a sale.
Gunners boss Arsene Wenger is having to contemplate changes to his squad, with Mesut Ozil and Alexis Sanchez having just six months remaining on their contracts and both Theo Walcott and Olivier Giroud being linked with moves away from the Emirates Stadium.
Real would need to sign a replacement should they give the green light for Benzema to leave the club, but Zinedine Zidane is under pressure to lead a revival during the second half of the season after overseeing the team's poor form on both the domestic and European scene.
Benzema has been at Madrid since 2009, scoring 175 goals in 344 appearances in all competitions.