However, Real Madrid will only accept a deal if Hector Bellerin is included as part of transfer which sees Benzema move to the Emirates.
According to Spanish outlet Diario Gol, Arsenal have been targeting the 30-year-old for a number of years.
But Arsene Wenger is not willing to spend more than €40million (£35.5m) on taking him from the Bernabeu.
Real president Flortentino Perez is thought to want at least a £20m raise to allow him to part with the France international.
That is unless Bellerin can be included in the deal.
Barcelona have been linked with a move for the defender in recent weeks as rumours of his discontent in north London begin to spread.
But Real were hoping to steal a march on their arch-rivals by opening negotiations with the Gunners.
Diario Gol add that Perez has other offers for Benzema, including an €80m (£71m) bid from China.
The Los Blancos chief apparently wants rid of the attacker in the summer as he has only scored four La Liga goals this term.
Real are expected to spend big on bringing in a new striker in the summer and Tottenham’s Harry Kane is top of their priority list.
Eden Hazard, of Chelsea, is also wanted by the Spanish outfit.