The two clubs are working on the final details of a deal to bring the Japanese international to La Real ahead of the 2022/23 season.
The Basque giants were initially interested in a loan move for the 21-year-old but they have stepped up negotiations on a permanent agreement.
However, as per reports from Marca, a breakthrough has been reached on a structured deal, involving contract until 2027.
Real Sociedad will pay an up front fee of €6m, but Real Madrid will keep a 50% ownership of his contractual rights, in the event of a sale during his time at the Estadio Anoeta.
Real Madrid will also have a buy back option on Kubo as Los Blancos keep an open mind on his long term future.