Muscat is on track for a second Chinese Super League title at Shanghai Port FC, who are on top by two points, with three league games to go.
The former Socceroos hardman is expected to finish that campaign, which ends on November 22, before departing for Rangers, where he won a domestic treble as a player in 2002-2003 and has previously been overlooked for the top job.
If appointed, Muscat, 52, will be the most high-profile Australian men's coach in Europe, following Ange Postecoglou's sacking at Nottingham Forest.
Rangers, who dismissed Russell Martin after just 123 days in charge, sit sixth, 13 points off leaders Hearts, and have already been turned down by Steven Gerrard and Danny Rohl.
The Rangers job appears a golden opportunity for Muscat to make his mark in Europe, having bounced back thoroughly from a short, ill-fated stint at Belgian club Sint-Truiden in 2019-20.
Throughout his coaching career, Muscat has consistently had to, and successfully managed to, step out from Postecoglou's shadow.
He succeeded Postecoglou first at Melbourne Victory, then later Yokohama F. Marinos.
Arguably unfairly, Muscat has had to shake off criticism that he inherited those teams from Postecoglou - despite enjoying success with a different game style and fresh recruits.
After finishing his playing days at the Victory, Muscat started his coaching career as Postecoglou's assistant there.
He took the reins after Postecoglou took the Socceroos job and carved out a coaching legacy at the Victory, winning two A-League grand finals, a premiership and an FFA Cup before quitting in May 2019.
After his short Belgian stint, Muscat was appointed manager of Marinos in July 2021, replacing now-Tottenham boss Postecoglou.
Muscat finished second in his first year at the helm in Japan, then won the league in 2022, before departing in December 2023 to make the switch to China.
Celtic boss Brendan Rodgers is prepared for what Muscat can do if he takes the helm at their fierce rivals.
"I don't know him that well, but he's done a really good job in the teams that he's been at in Asia and if he's to be the new manager coming in, it's another new challenge and ... we'll be ready for it," he said.
Muscat was sounded out for the Socceroos head coach role following Graham Arnold's departure, but chose to focus on his club football, with Tony Popovic ultimately appointed.
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