Club president Josu Urrutia and Bielsa left the meeting without speaking to reporters, failing to dampen Spanish media speculation that the eccentric and outspoken Argentine was considering his position.
The problem blew up at the beginning of the week when the team returned to start pre-season training after the holidays, and Bielsa lost his temper with contractors carrying out renovation work at the Lezama training facilities.
The Argentine gave a lengthy news conference on Thursday when he apologised for the way in which he had acted, but insisted the work was shoddy and behind schedule and that he was only defending the club's interests.
He accused the contractors of "ripping-off, robbing, and deceiving" the club, but Bilbao's board responded on Friday with a strongly-worded statement on their website saying they completely disagreed with their coach on all counts.
"(The club) apologise publicly for the damage that has been done to the construction company and their workers by an employee of Athletic Bilbao," the Basque side said.
The former Argentina and Chile boss joined Bilbao at the beginning of last season guiding them to the finals of the Europa League and Copa del Rey, only to lose both to Atletico Madrid and Barcelona respectively.