The striker is a long term target of Arsene Wenger and the club have made an early move for the 21-year-old, who is also wanted by Juventus.
Madrid are willing to sell, but will insist on a buy-back clause which is likely to prove a key stumbling block towards a deal.
Arsenal tried to sign the Spaniard on loan last summer and Arsenal hope the lure of regular football can help secure the move.
Morata scored nine goals in 28 games for Real this season, and came off the bench in Saturday’s Champions League final to help them to victory against Athletico Madrid.
The Gunners are also interested in Real striker Karim Benzema but Carlo Ancelotti wants the France international to sign a new contract.