The Greece-born Australian coach is the new frontrunner to replace Neil Lennon at Parkhead following the collapse of talks with former Bournemouth boss Eddie Howe last week.
Postecoglou is currently in charge of J-League side Yokohama F Marinos, who he guided to the Japanese title for the first time in 15 years in 2019.
The 55-year-old also has an Australian championship on his CV having won the A-League title with Brisbane Roar in 2011, while he also led Australia to the World Cup in Brazil in 2014 as well as guiding the nation to the Asian Cup title the following year.
His success with club and country has attracted the interest of Celtic, but it is claimed that the move could be 'in jeapordy' due to a coaching licence issue.
European football guidelines state that any top flight manager must hold UEFA A and B licences - or have started a UEFA Pro Licence course - but Postecoglou does not have the necessary qualifications, the Times has reported.
His only experience of coaching in Europe came during a short stint at Panachaiki in the Greek third division in 2008, where a reported fall-out with club management ended his tenure.
Postecoglou was coy when asked about his future following his side’s late 2-1 win over Shimizu S-Pulse at the weekend. "I’ve understood there’s some speculation around the future, but I’ve stopped thinking about the what-ifs in the world a long time ago," he said.
"My focus the last couple of days has been on making sure we [Marinos] play well."