Having announced Postecoglou as their manager earlier this month, Celtic have now turned their attention towards bolstering their squad ahead of the new season.
The Australian tactician is said to be keen to strengthen his midfield options and the Bhoys appear to have a number of potential targets in that position.
Mooy, who has plied his trade under Postecoglou at international level, has been linked with a move to Parkhead, with the Celtic boss keen for the club to pursue a deal for him.
Postecoglou is said to have asked Celtic’s recruitment team to prioritise the Australia international as a midfield target over other potential options.
The Hoops have been claimed to be looking into a deal for Ireland international McCarthy, who is set to become a free agent when his contract with Crystal Palace comes to an end this week.
However, the Glasgow giants could put their attempts to sign the Irishman on hold to prioritise a deal for Mooy at the request of their new manager.
Mooy is currently playing in China for Shanghai SIPG and it remains to be seen if Celtic can strike a deal with the club for the midfielder.