Sociedad have won just once in the league all season and, after Sunday's defeat to Espanyol, sit 15th in La Liga following a promising campaign under Arrasate last time out.
In his first full season in charge, Arrasate guided Sociedad to seventh in the top flight and the UEFA Europa League's qualification stages.
However, elimination from continental competition at the hands of Krasnodar set the tone for a poor start to their league campaign.
Despite struggling early on, president Aperribay warned the club will not be making any rash decisions on their coach's future.
"Having five points is not nice and is not the place where we want to be. It is difficult to know what the causes are," he told Mundo Deportivo.
"This is a war in which we must roll up our sleeves and improve.
"It is not a matter of making gestures, we need to analyse and think what to do to fix it. We have to analyse and talk with the coach, the directors and the players.
"It is not an issue of not trusting the coach. We have absolute confidence. There are lots of things that can be done and must be improved but we have to talk to those involved and improve the team's situation
"The dressing room was deeply depressed [after the loss to Espanyol]. The attitude was to try and win but nothing has come."
