The Spain Under-21 international is reported to be on the verge of completing a €22million (£17.5m) move to the Italian champions.
Morata made his debut for Real back in December 2010 and was part of the first team under coach Carlo Ancelotti last season.
However, the 21-year-old has decided to leave the Santiago Bernabeu in order to play regular football and will reportedly sign a four-year contract with Juve, with Real inserting a buy-back clause that will be activated in 2016.
Morata is delighted to be joining the Turin club, who have won Serie A for the past three seasons.
"I'm very happy,” Marca quotes the youngster as saying. "This is a new challenge for me."