Widespread reports this week said Allardyce will leave West Ham when his contract expires at the end of the season.
But Big Sam insists he is the right man to lead the club, particularly if they qualify - as is expected - for the Europa League through the FIFA Fair Play route.
Yesterday reports claimed Besiktas boss and former Hammers hero Slaven Bilic and Napoli manager Rafa Benitez were in a straight shootout to replace the former Bolton and Blackburn manager in the Upton Park hotseat.
But Bielsa has come back into the frame from left field after admitting his future in France is unclear.
The Argentine said he is still waiting to see if he will be offered a contract for next season with his side currently fourth in Ligue 1 with only an outside chance of finishing third and claiming the last Champions League spot.
Speaking at a news conference on Thursday, as reported by the Daily Mail, Bielsa said: "If I receive a concrete offer, I will read it as any coach would and I'll give an answer."
The 59-year-old has plenty of experience. He used to manage both Argentina and Chile and could excite fans as he is something of a footballing eccentric who is famous for deploying a 3-3-3-1 formation with an aggressive pressing style.
His stats are pretty impressive too. He boasts a winning percentage of 48.58 in his career so far, with 137 victories taken from 282 games as a manager.
While some fans are eager for a different approach and style to what Allardyce brings to East London, some feel Bielsa is perhaps too much of an extreme change.
Known as a strict disciplinarian, his preference for all out attack has endeared him to Marseille fans, but he has also been criticised at times for being too inflexible and naive.
Both co-owners David Sullivan and David Gold and Allardyce have always insisted they will sit down and discuss the future at the end of the season.
Whatever the decision, the wait is almost over with two games and little more than a week left until judgement day.