This is according to the Daily Mail, who understand the Gunners are prepared to pay £45 million for the France international, plus £150,000-a-week in wages.
Wenger, the report adds, is desperate to sign the 27-year-old - believing the striker is the man required to turn Arsenal into genuine title contenders this season.
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Twice in the past seven days, Wenger has publicly denied his interest in Benzema, one of the world’s most complete centre-forwards.
But it’s understood Wenger is secretly hopeful that a huge offer - worth in excess of the £42.5 million Arsenal paid Real Madrid for Mesut Ozil two summers ago - would convince the Spanish giants to do business.
Real Madrid want a replacement - preferably Aguero
Benzema, if he signs for the Gunners, would become the highest-paid player at the Emirates Stadium.
However, one potential problem is the fact Real Madrid would ideally want to sign a top-level replacement for Benzema before allowing the former Lyon star to leave the Bernabeu.
Sergio Aguero would reportedly be their first choice replacement, but there is no chance of Manchester City letting the Argentina international go.
Benzema could turn Arsenal into champions
Ahead of Arsenal’s Premier League opener against West Ham on Sunday afternoon, Wenger told reporters he was happy with his options upfront, naming Alexis Sanchez and Danny Welbeck as two of his forwards who, like Theo Walcott and Olivier Giroud, are capable of playing upfront.
However, there’s no doubt that Benzema would be an upgrade on all four aforementioned forwards.
If Arsenal manage to sign the unselfish striker who has scored 133 goals in 281 appearances for Real Madrid over the past six years then their chances of winning their first Premier League title since 2004 will increase significantly.
Arsenal fans: would you win the Premier League title this season with Karim Benzema in the side? Have your say by leaving a comment in the box below!