Gazzetta dello Sport and Turin-based sports newspaper Tuttosport claim the Spain Under-21 international is set to join the Italian champions on a £17.5million transfer deal .
Morata, who was part of Real's first team under coach Carlo Ancelotti last season, is set to sign a four-year contract with the Bianconeri outfit.
However, the 21-year-old is keen to leave the Santiago Bernabeu in search of more playing time.
Morata made just three La Liga starts in the 2013-14 campaign and has scored eight goals in 23 league appearances overall.
He has one year left in his contract with the European champions, and is also a reported target of Arsenal and Wolfsburg.