The midfielder will join Real Madrid Castilla, the club's reserve team.
Madrid moved in ahead of Barcelona by signing Kubo, as the Catalans had him in their ranks as a youth but was one of the players that led to their 2014 Federation Internationale de Football Association sanction and he returned to Japan.
Kubo apparently demanded an annual salary of €1 million (R16.7 million) for each of the five years of his contract with the Blaugrana, in addition to wanting to be guaranteed promotion to the first team in his second season with the club.
Los Blancos have offered Kubo two million euros a year to play in Real's second-string Castilla side to gain experience, according to Japanese media.
It will not be Kubo's first taste of Spain, however, as the young prospect was invited to enter the Barcelona youth academy at age 9, earning him the nickname "Japanese Messi".
Barcelona have kept tabs on the player since, but Real Madrid now look to be the front-runners to sign Kubo, whose dribbling skills made him a YouTube hit as a small boy.
Turning 18 has meant he can officially move back to Europe.
The agreement comes days after the youngster made his for senior appearance for Japan in a pre-Copa America friendly against El Salvador.
Los Blancos have already spent big this summer signing Eden Hazard, Luka Jovic, Ferland Mendy and Eder Militao and could yet add more.