The 19-year-old Japan international is yet to feature for Madrid's first team in a competitive fixture, having signed last year before sealing a temporary switch to Real Mallorca.
Kubo played 35 times in LaLiga this season, scoring four goals.
The former FC Tokyo prospect could soon be joined by further new arrivals, meanwhile, as Villarreal reportedly close on a deal for Dani Parejo and Francis Coquelin.
Valencia skipper Parejo last month said he wanted to retire with Los Che despite talk of a departure, although his captaincy was subsequently criticised by the outgoing Ferran Torres.
It is said Villarreal would pay rivals Valencia €13million for the midfield pair.