Four goals from Morata at the finals in Israel helped Spain win the tournament, and earned the 20-year-old the Golden Boot.
But just five starts for Real last season had provoked speculation that the striker may seek a move away from the Bernabeu.
However his success with his country's Under 21s has made Morata keen to make his mark when Spanish domestic football resumes.
Morata told Cadena SER: "I called the president and he told me that I will stay at Real Madrid next season. He is very happy with me."
As well as pledging his future to Real, Morata is hopeful that Under 21 team-mates Isco and Thiago Alcantara may join him from Malaga and Barcelona respectively.
"I am trying to convince Isco to come. It seems Thiago has received bids, and they are enough to make him consider leaving (Barca)."