That is according to Spanish source Diario Gol, who have given just about everyone reason to question their transfer claims in the past. But per their report, the Gunners recently had their people meet with the Frenchman's agent at a Madrid hotel.
They are also claiming that the London side are prepared to offer £52.4m (€60m) to secure a deal for the player.
Benzema is said to be upset over his exclusion from France's World Cup squad. And after scoring just 11 goals during the season, Los Blancos are also looking to offload him.
Real president Florentino Perez is rumoured to be keen on a deal for Bayern Munich and Poland forward Robert Lewandowski. But the club are also heavily linked with Paris Saint-Germain star Neymar.
Arsenal, though, are understood to be facing a lot of competition for the 30-year-old, with Serie A outfits AC Milan and Inter also having sent their representatives to Spain for negotiations.
Benzema, who is contracted to Madrid until 2021, has already rejected a lucrative move to China as he wants to keep playing high-level football, and it will be interesting to see what unfolds over the summer.
From a realistic standpoint, Arsenal spending over £50m on a 30-year-old forward who has had a poor season, especially when they already have both Alexandre Lacazette and Pierre-Emerick Aubameyang in attack, just doesn't seem likely.
They are also in a transitional period and are yet to announce a manager, something which could leave transfer negotiations on hold for a while.