The 21-year-old impressed with his performances for Zinedine Zidane's side in the Supercopa de Espana earlier this month but is still behind the likes of Gareth Bale and Isco in the pecking order for La Liga.
Real have opened talks with the youngster over a new deal, but Arsenal are said to be hoping to take advantage of his relatively low release clause by making an offer before he commits to fresh terms.
According to Diario Gol, Wenger is expected to offer Asensio the promise of regular first-team football if he moves to the Emirates this summer. Madrid, meanwhile, were reported "shook" by Arsenal's interest and have called for "emergency talks" to thrash out a new deal.
Asensio moved to the Bernabeu from Mallorca in 2014 and has so far made just 24 La Liga appearances for Los Blancos.