The Gunners are believed to be one of a number of clubs circling the talented Spaniard, who has emerged as one of his nation’s brightest talents in recent times.
Some rumours suggest that the London club have already sounded out the 21-year-old ahead of a summer raid and that personal terms between both parties have been agreed.
It is thought that only a fee has to be settled with Real to see the deal completed.
The La Liga giants value their man highly, and are said to be demanding £25m to let him depart.
Despite their wealth, the Madrid side are relatively short on central-attacking options, with first choice man Karim Benzema only backed up by Morata and Jese Rodriguez – another young player.
Such a fee may deter Arsene Wenger, but it is understood that Real will not push to have a buy-back clause inserted.
Morata starred at last year’s U-21 European Championships, where he finished the tournament as top scorer with four goals as La Rojita lifted the trophy.