Bielsa had indicated his intention to resign on Friday after the club refused to back his critical statements of the construction company employed to improve the facilities at the club's Lezama training base.
However, after meeting with the club's board, Bielsa was convinced to rethink his decision and continue for at least another trial period of two weeks.
The 56-year-old had railed against the work currently taking place at Lezama in a press conference on Thursday, describing it as "like a fraud or a robbery".
The club were quick to distance themselves from those comments as they issued a statement of their own on Friday morning saying they disagreed entirely with Bielsa's sentiments and apologised to the company for the public criticism on behalf of one of their employees.
The former Argentina boss was then reported to have handed in his resignation, but after a three-hour meeting with the board where his comments were fully explained and discussed, Bielsa was convinced by president Josu Urrutia to delay any decision until after a period of reflection.
Bielsa enjoyed a successful first season in charge of the Basque club as he guided them to their first European final in 35 years and also to the Copa del Rey final, where they were beaten 3-0 on both occasions by Atletico Madrid and Barcelona respectively.