Birighitti was on the wrong end of a 4-2 defeat to Postecoglou's Celtic at Parkhead on Saturday but produced a number of good saves that saw his fellow countryman offer him some words of encouragement afterwards.
Postecoglou was in charge of the Socceroos when Birighitti earned his only senior cap to date in a 4-3 defeat to China in the 2013 EAFF East Asian Cup, however, he missed out on his final selection for the 2014 World Cup in Brazil.
Australia are due to name their squad for the 2022 tournament in Qatar on Monday and the United stopper is still hopeful of making the cut, having moved to the Scottish Premiership in the summer in order to enhance his chances.
"Fingers crossed," he said. "To get selected for the national team on any occasion let alone a World Cup is a dream come true. That is what any kid growing up back home dreams about – to represent their country. I have been lucky enough to do it once and be capped for the national team.
"If I get a call up I will be buzzing. But if that is not the case then I will keep working hard day in day out and trying to be the best version of myself. I will try to get better each day and put in performances which can help the team. It is going to be needed. Hopefully we can turn it around on Wednesday night at home [against Kilmarnock].”
Birighitti also spoke of his admiration for Postecoglou, revealing that the Celtic boss took time out to speak to him after Saturday's defeat at Parkhead.
“He just said that I had played well and told me to keep my head up and stay positive and that results would turn. It was good of him. Ange is a class character on and off the field.
"I have the utmost respect for him and what he is doing. He has done unbelievably well and I can’t give him enough credit. There are not enough words to describe what a man he is.”