The Australian arrived in Glasgow today having been required to quarantine in London following his arrival from Asia after leaving Japanese side Yokohama F. Marinos to take up the Parkhead post.
Postecoglou met with the club’s new chief executive Dom McKay in London to lay groundwork for the new season.
The 55-year-old also took in Scotland's Euro 2020 clash with England at Wembley on Friday night.
Celtic have already returned to training and will head to a training camp in Wales on July 5 ahead of their first pre-season friendly against Sheffield Wednesday on July 7.
It will be one of four warm-up matches before the UEFA Champions League second qualifying round tie with Danish side FC Midtjylland at Celtic Park on Tuesday, July 20.
One bit of earlier business could be a contract extension for Leigh Griffiths who revealed he has spoken to Postecoglou.