Kubo, who previously played at Barcelona's La Masia academy, joined Madrid from J.League outfit Tokyo for a €2million fee last month.
The highly rated Japan international is set to play for Madrid's reserve team Castilla in 2019-20, but the 18-year-old has made an immediate impression during the club's pre-season training camp in Montreal.
Discussing his start to life with Madrid, Kubo – who spent four years at Barca before leaving in 2015 after the Catalan club were found to have violated FIFA's international transfer policy for under-18 youths – told Real Madrid TV: "In Japan there is a lot of passion for Real Madrid and I am very proud to play for the best club in the world.
"Players and coaches here are very friendly and I have been able to adapt very well, we have to take advantage of this opportunity, but you also have to have fun because not everyone has this opportunity.
"I'm learning a lot as a footballer and as a person, both on and off the field, I'm having a great time."
Madrid kick-off their International Champions Cup campaign against Bundesliga holders Bayern Munich in Texas on Saturday.