The Hoops enjoyed two successful years under the tutelage of Postecoglou, winning the domestic treble in 2022-23, but the Australian has decided to seek pastures new and take the top job at Premier League outfit Tottenham Hotspur.
Rodgers has emerged as one of the early frontrunners to potentially replace Postecoglou, and he is currently out of work following his exit from Leicester City in April.
The 50-year-old won both the FA Cup and Community Shield during a four-year spell in charge of the Foxes and he also led the club into Europe, reaching the Europa Conference League semi-finals in 2021-22.
Rodgers endured a difficult campaign with Leicester this term, though, and was unable to steer the club away from the Premier League relegation zone - the Foxes eventually finished 18th in the table under interim boss Dean Smith and will play Championship football next season.
Before joining Leicester, Rodgers spent three successful years as Celtic manager between 2016 and 2019, winning seven domestic trophies including back-to-back trebles in his first two seasons at Parkhead.
The former Liverpool boss won 69.8% of his 169 games in charge of the Glaswegian giants, lifting two Premiership titles, two Scottish Cups and three League Cups, before ending his reign in February 2019 to join Leicester.
According to The Mail, Rodgers retains the support of Celtic's major shareholder Dermot Desmond, and the former Hoops boss recently held face-to-face talks with chief executive Michael Nicholson and finance director Chris McKay over a possible return to Parkhead.
The report adds that both Nicholson and McKay took a private jet to meet Rodgers in his home in Mallorca on Sunday.
The Northern Irishman is said to have initially had plans to take a sabbatical from management following his departure from Leicester, but he is now "weighing up the pros and cons" of a second spell in charge of Celtic.
Celtic supporters have mixed views over the prospect of Rodgers returning to the club following his mid-season exit to Leicester four years ago.
The report also states that the potential arrival of Rodgers could persuade Celtic assistant coach John Kennedy to remain at the club.
Kennedy worked under both Rodgers and Postecoglou and is said to be wanted by the latter at Tottenham.
Celtic have also been linked with a number of other managerial candidates including Manchester City assistant coach Enzo Maresca, with a recent report claiming that they have intensified their pursuit of the 43-year-old Italian.
The likes of former Leeds United head coach Jesse Marsch, Bodo/Glimt boss Kjetil Knutsen and former Borussia Monchengladbach manager Daniel Farke are also thought to be under consideration.