Reports have suggested that Nick Montgomery could be in line for the role vacated by Chris Davies in north London.
The ex-Hoops coach recently left his position working under Postecoglou in order to become the head coach of Birmingham City in League One.
FTBL states Postecoglou has now turned his attention to former Hibernian boss Montgomery as his successor.
The 42-year-old was sacked by chiefs at Easter Road towards the end of last season after the club's failure to finish in the top six of the division.
He was permanently replaced by Scottish Cup-winning hero David Gray, who is working under new sporting director Malky Mackay.
Postecoglou will be familiar with Montgomery's body of work carried out in Australia before his experience in Scotland.
He had an excellent record in the A-League, with a title triumph at Central Coast Mariners his biggest achievement.
The former midfielder was lined up for a potential return to management a month ago with Western Sydney Wanderers but instead they brought in Perth Glory's Alen Stajcic.
"It was an honour to be appointed manager of Hibernian Football Club and to become part of a club with so much history," Monty said after leaving Hibs. "After arriving with my Assistant Manager Sergio (Raimundo) and Goalkeeping Coach Miguel (Miranda), at a time when the club had been through a difficult start to the season, I was confident that we could lift the club and help to challenge for both cups and the top six positions.
“Whilst I am proud we made the semi-final of the League Cup and the quarter-final of the Scottish Cup, I am disappointed that we narrowly missed out of the top six in the last minute of the final game.
“I am immensely proud to have given a debut to the youngest player in the club's history, as well as four other academy graduates, and I hope they will continue their development to become first team regulars in the near future.
"I depart having met some wonderful people and witnessed the passionate fan base who showed support throughout all of the challenges the club faced during the season."