"I think his decision to leave was taken before (last) Wednesday", said Labrune at a press conference on Thursday.
Bielsa stunned French football by quitting OM immediately after his side began their season with a 1-0 defeat at home to Caen last Saturday.
His departure followed weeks of uncertainty over whether he would agree new terms to ensure he stayed at Marseille for a second season, but the club appeared to be completely unprepared for the announcement.
Labrune added that the press conference was "absolutely not" organised in order "to give the name of the next coach".
Former Borussia Dortmund coach Jurgen Klopp has already ruled himself out of the running to succeed Bielsa, who led OM to fourth place in Ligue 1 last season.
The Marseille president said the club had "full confidence" in interim coach Franck Passi. And while he admitted to being "stunned" and "disappointed' by Bielsa's decision to quit, he added: "We knew that Marcelo Bielsa was unpredictable. We had full knowledge of the facts and I take full responsibility."