The club have been actively pursuing the young talent for a few weeks now and according to Gazzetta Dello Sport, President Erick Thohir has landed in Madrid in order to sign the 21 year-old.
It is believed that Inter Milan want to make the move for Morata on loan with an option to purchase at a price of around 22 million euros.
Real Madrid would also have a buy-back clause set at 25 million euros, much like Dani Carvajal's move which saw him move to Bayer Leverkusen before being bought back after just one season.
But the Under-21 international could instead opt to sign a renewal with the Spanish giants which would most certainly end the speculation surrounding him where the likes of Juventus and Arsenal are also showing interest.
Morata has made just one start in his 18 league appearances this season but has managed to score six goals which is an extremely good return for someone who has been on the pitch for less than 300 minutes.