But the Palma de Mallorca-born player is now back at Real and was unveiled to the media on Tuesday.
He said in a news conference: "I don't fell vertigo, I'm just enthusiastic and eager to do the best I can."
Asensio has taken full advantage of the opportunities handed to him by coach Zinedine Zidane during pre-season to show what he is capable of.
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"Zidane has always shown confidence in me," he said. "He told me that he was happy, that he wanted me to stay and I also wanted that.
"He has told me he will be fair but that I would have to earn my minutes in training."
Asensio played the full 120 minutes in last week's UEFA Super Cup final against Sevilla, scoring Real's opener in a 3-2 triumph, his first official game for the club.
"I'm delighted," he said. "It was unforgettable to win the Super Cup and score. Now I have to continue to work hard to earn more playing time."
Real, who finished runners-up to Barcelona in La Liga last season, have not won Spain's top flight since the 2011-12 campaign.
Asensio cannot wait to bring more trophies to0 the club.
"I'm very ambitious," he said. "I want to win it all."
Club president Florentino Perez said Asensio, who is under contract until June 2022, has a bright future at the club.
"We welcome a marvellous player," he said. "His quality is undisputed."
He then turned to an emotional Asensio and said: "Twelve years ago your parents met me in Mallorca and told me that you, their son, would one day play for Real.
"You were eight years old then.
"Real Madrid has chosen you to be a part of our history and you have chosen us."
(Reporting by Adriana Garcia, editing by Neil Robinson)
