Although he has yet to make a formal transfer request, Bale is thought to be frustrated by Spurs' refusal to sell him, while the English club are said to be holding out for in incredible 100 million pounds.
He joins fellow under-21 teammates Francisco "Isco" Alarcon and Daniel Carvajal as Madrid's reinforcements this offseason for new coach Carlo Ancelotti.
The 23-year-old Illarramendi had a breakout season in the last campaign when he helped Sociedad finish fourth in the Spanish league and earn a Champions League berth.