The 24-year-old has been a long-term target for Gunners manager Arsene Wenger, who failed with a bid to sign him from Juventus last summer.
Morata eventually moved back to Madrid after the La Liga side exercised a €30m (£26.2m) buy-back clause in his contract and he has since gone on to make 26 appearances for Los Blancos as part of their remarkable 40-game unbeaten streak.
According to OK Diario, Wenger thinks that Morata would make a good partner for Olivier Giroud, who penned a new deal at the Emirates yesterday, and is preparing an "irresistible offer" for his services this month.
The Gunners would likely have to shatter the club-record £38.2m they paid for Mesut Ozil in order to land the Spaniard, who was reportedly subject of a rejected €75m (£65.5m) bid from Chelsea last year.
Morata has registered nine goals and five assists in all competitions for Zinedine Zidane's side so far this season.