Asensio is the son of a Spanish father and a Dutch mother. He moved on in 2013 from the youth of RCD Mallorca. For this, he made his debut in the first team on October 27, 2013, in a match in the Segunda Division against Recreativo Huelva. Coach José Luis Oltra transferred Asensio to the selection of the main force after a few games. Asensio started on 2 February 2014 for the first time in the starting eleven, against Sporting Gijón. He made his first goal in professional football on March 16, 2014, in a league match against CD Tenerife.
Asensio signed a contract until 2021 with Real Madrid in December 2014. That left him with RCD Mallorca until the end of the season and rented it to RCD Espanyol during 2015/16. Before that he played in the Primera Division for the first time. Asensio made its official debut in the first team of Real Madrid on August 21, 2016. That day he played as left winger during a league match won 0-3 at Real Sociedad. He made the 0–2 in the 40th minute.