The Japan international signed for Real Madrid in 2019, but since arriving at the club he has failed to make a single appearance for the senior side.
Instead of featuring for Los Blancos, Kubo has spent the last three seasons on loan at three different La Liga clubs, including two spells with Mallorca.
After experiencing four loan spells over the last few years, Kubo is set to leave Real Madrid on a permanent deal in the current transfer window.
According to Marca, Real Sociedad are closing in on a transfer for the 21-year-old after initially rejecting the chance to sign Kubo on a loan deal.
The Basque side were reportedly keen on bringing Kubo to San Sebastian on a permanent basis, with the player set to sign a five-year contract with the club.
Kubo is reportedly attracted by the chance to play in the Europa League next season and to join a team that plays an attacking brand of football.
The winger scored two goals and contributed three assists in 30 appearances for Mallorca last term.