The former Chelsea boss is widely expected to leave the Spanish capital at the end of the season after a poor start to the La Liga season and rumours of dressing room disputes.
This has caused Galacticos to search the market for a new manager, with Wenger being identified as a top target.
The Gunners chief has turned down opportunities to manage at the Bernabeu in the past, but the club’s president Florentino Perez is confident that he can convince the Frenchman to make a move this summer.
Arsenal have struggled in the Premier League so far this term, leading to criticism of Wenger’s managerial abilities and approach in the transfer market.
As well as current issues, there remains the club’s eight-year trophy drought, which has frustrated fans at the Emirates Stadium.
Real are willing to match Wenger’s £7m-per-year salary in a bid to sway his decision.
Perez is running for re-election at Real and a solid managerial appointment could boost his chances of keeping his position.
Chelsea’s interim manager Rafael Benitez may be another, and could be keen to return to Spain.