It would be a hugely popular move with fans who thrilled to his style of play and success when he managed the club from 2021-23.
The Australian enjoyed two trophy-laden seasons at Celtic Park before heading south to manage Tottenham in the English Premier League.
Sacked by Spurs despite winning their first trophy in 17 years, the Europa League, he subsequently had a brief unsuccessful spell at Nottingham Forest and is currently out of work.
Scottish bookmakers priced Postecoglou at 4-1 for a return to Celtic. He is third favourite behind former player Shaun Maloney and Martin O'Neill, another ex-Bhoys boss who had a successful caretaker stint, with Maloney alongside, before Nancy's arrival.
Postecoglou quickly distanced himself from a return after Celtic parted company with Brendan Rodgers in October, but that was barely a week after his own departure from Forest.
Now, having had a break, he may be tempted - or he might prefer to take over at the end of the season. That would enable him to have a full pre-season and a transfer clear-out, with Celtic potentially returning to O'Neill as caretaker in the interim.
The slate is clean at Celtic with Nancy's three assistants also fired and head of football operations Paul Tisdale exiting too in the wake of Saturday's 3-1 home defeat by bitter rivals Rangers.
In addition Peter Lawwell stepped down as club chairman last month blaming increasing abuse and threats which had become 'intolerable'.
In their statement, which said the club had "decided to terminate the contract of manager Wilfried Nancy with immediate effect" Celtic added: "A further update will be provided to supporters as soon as is practical".
Celtic began the season under Rodgers, who had led them to the title in the past two seasons, but he quit in October after poor form and disagreements with the board.
O'Neill then took over as caretaker stabilising results. Under the 73-year-old Celtic won seven of eight matches and drew level on points in the league with Hearts - having been six points adrift when he arrived.
Nancy, who had previously coached Columbus Crew in MLS, winning two cups, replaced O'Neill last month but Celtic failed to keep a clean sheet under the 48-year-old Frenchman and are again six points behind Hearts.
Nancy's 33-day reign was shorter than O'Neill's caretaker stint by four days.
Celtic play Dundee United on Saturday, starting a hectic run of eight games in 24 days including a trip to Hearts on January 25.
With PA
39°C