The Arsenal legend has been left out after a run of one win in 14 Premier League matches. The Eagles are just three points above the relegation zone.
Now the bookmakers have already opened bets on the next manager of Palace and Postecoglou is among the favourites.
Celtic manager at 9/1, with Rafa Benitez at 6/1 and Nuno Espirito Santo at 7/1.
Postecoglou will have caught the eye after his stunning conversion at Celtic and the club are currently on course to claim a treble this season.
Speaking recently about the speculation linking him to jobs in the South, Postecoglou said: “It’s fair to say that speculation about my future has been with me since day one, you know, for various reasons!
“For me, I don’t worry about that stuff. It just doesn’t really enter my brain space.”
“I do every job I have and every challenge to try to make the football team as successful as possible and that’s always been my complete focus. If I started to think about the speculation that is out there now and possibly moving to I don’t know where I would be worried initially when people were saying I wasn’t going to last long .
“I keep telling players, just live in the moment. The present is the most important time because the past, you can’t change and the future, you can’t control.”
Former Rangers boss Steven Gerrard is also listed by the bookies as potential candidates and priced at 12/1.
Palace chairman Steve Parish told the club’s website: “This difficult decision has been taken with great regret.
“Ultimately, results in recent months have put us in a precarious position in the league and we felt that change was necessary to give us the best chance of retaining Premier League status.
However, Patrick’s influence since joining us in the summer of 2021 has been significant, and he commands great respect from myself and all of his colleagues.
“He led the team to the semi-finals of the Wembley Cup and finished 12th last season playing some exciting football, which has been a difficult and crucial campaign for the club considering the changes we made to the squad prior to his arrival.”
Three members of Vieira’s coaching staff – Ossien Roberts, Christian Wilson and Said Aegon – also left Selhurst Park.
Paris added: “Patrick has done everything he can for the club and we thank all of him and his team for their service.
“We also wish Patrick, Ossien, Christian and Saïd all the best for their future.”
Palace say the process of appointing a new manager is ongoing and they will provide an update in due course. Dean Kelly will remain as goalkeeping coach.
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