Bayer are seeking a new coaching appointment after dismissing Erik ten Hag just three games into the new Bundesliga season.
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Transfer expert Fabrizio Romano is reporting Australian Postecoglou is being discussed inside the Bayer boardroom.
Postecoglou is available after being sacked by Tottenham barely a week after leading them to the Europa League title - a first European trophy for Spurs in 45 years.
Along with Bayer, Fenerbahce are also interested in Postecolgou after this week's dismissal by Jose Mourinho.