Greek-born Postecoglou, who has also taken charge of Brisbane Roar, Melbourne Victory, and the Australian national team, is priced at 1/1 with SkyBet and is the current favourite following the Howe bombshell.
Scotland boss Steve Clarke, former Nice and Borussia Dortmund manager Lucien Favre, and Hibs head coach Jack Ross also feature among the possible new managers.
1. Ange Postecoglou - 1/1
Currently with Yokohama F Marinos in Japan, the Greek-born coach has had spells managing Brisbane Roar, Melbourne Victory, and the Australia national team
Photo: Matt Roberts
2. Steve Clarke - 4/1
The Scotland manager has the small matter of Euro 2020 in front of him at the moment but is currently second favourite to succeed Neil Lennon
Photo: Ian MacNicol
3. Lucien Favre - 11/2
Swiss coach has managed some big teams including Hertha Berlin, Borussia Monchengladbach, Nice, and Borussia Dortmund
Photo: Lars Baron
4. Jack Ross - 13/2
Hibs boss led his side to a first third-pace finish since 2005 and the last four of both cup competitions in his first full season in charge
Photo: Rob Casey - SNS Group / SFA