Aussie coach Postecoglou has many admirers in Glasgow, where he enjoyed two successful seasons, winning five trophies, before heading to the English Premier League.
But on a day of mixed messages, it appears that Ipswich Town manager Kieran McKenna may have edged ahead of Postecoglou to become the bookies' favourite, while other reports suggested sources close to the Australian have 'played down' a return to Parkhead.
Meanwhile, third favourite Shaun Maloney, appointed caretaker alongside Martin O'Neill, has ruled himself out of the running.
Maloney, temporarily promoted from his role as player pathways manager, said he was "nowhere near anything short-term or long-term at the moment".
He added: "So at this present time, no. I'm going to try my best to support Martin and the players but I'm very similar to Martin, to be interim and then I'll go back to my role."
Maloney spoke as he prepared the Celtic players for Wednesday's visit of Falkirk, which follows successive league defeats and Monday's departure of Rodgers following a rift with principal shareholder Dermot Desmond.
Maloney said: "It was a turbulent day, traumatic day really, for everyone, not just at the stadium but the training ground. So, as expected, it was a tough morning."
O'Neill, 73, has also stressed that his involvement with Celtic can only be temporary and even joked that "if we don't win on Wednesday evening, we might have a new manager on Thursday".
Furious Celtic fans will protest against Desmond and the club's board before the Falkirk match and many want Postecoglou back at the helm.
The BBC carried social media comments such as: "Should be Ange back to get them going forward instead of sideways, backwards, etc" and "Get Ange back!!!".
But it is also possible that Postecoglou will want to have a break before venturing back into management.
Postecoglou left Scotland in 2023 to join Tottenham but despite winning the Europa League, lost his job in north London at the end of last season.
The 60-year-old was recently sacked from his second Premier League post with Nottingham Forest after eight winless games.
Rodgers' second spell at Celtic ended after Sunday's 3-1 Scottish Premiership defeat at Hearts left them eight points adrift of the Edinburgh side.
Desmond later released an explosive statement slamming Rodgers and suggesting his recent words and actions had been "divisive, misleading, and self-serving".
Rodgers is yet to respond to the statement.
