According to Sky Sports, Postecoglou and the Scottish football giant have “verbally agreed personal terms."
The 55-year-old is currently team manager of Japanese football club Yokohama F. Marinos.
SEN broadcaster Andy Maher and The Scotsman football journalist Barry Anderson reported that Postecoglou would lead Yokohama F. Marinos for a final time on Wednesday before jetting to Glasgow after the match.
The former head coach of the Australia national football team emerged as Celtic’s No. 1 target after negotiations with former Bournemouth boss Eddie Howe fell through.
Former Celtic player and multiple Premier League club manager Paul Lambert predicted Postecoglou would fit into the club promptly.
“If that’s the case, it’s great news for everybody in Glasgow, the Celtic supporters especially,” Lambert told Sky Sports.
“It’s been a long time, but hopefully they’ve got the right guy and he goes and rescues the title back.
“I think it’s important he puts his own mark on it, which I’m pretty sure he will do. He’ll find his feet pretty quickly.”
Reports last week suggested Postecoglou could bring in former Socceroo and Melbourne Victory coach, Kevin Muscat, as his assistant at Celtic.
Postecoglou won the Asian Cup with Australia in 2015 and the J-League Title with Yokohama FM (his current club) in 2019.