The former Gers fullback is now favourite for the job, says BBC Sport, with negotiations at an advanced stage.
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Muscat, should he take the job, would move to Ibrox in late November after the Chinese League ends.
The former Australia captain has won league titles as a manager in Australia with Melbourne Victory, Japan with Yokohama F Marinos and China with his current club.
With Rangers, Muscat won the domestic Treble in 2002/03.