It is understood that construction of the new San Sebastian, which will also bring the removal of the athletics track, will begin in 2015.
"Execution of the new Anoeta project will consist of four phases, taking place over two seasons. The current budget we are managing is €48.3m (£39.5m)," Inside Spanish Football quotes Aperribay as saying.
"The final capacity will permit 42,300 spectators."
Sociedad finished fourth in La Liga last season, but currently sit sixth in the table this time around, nine points behind fourth-placed Athletic Bilbao with six games remaining.