Michael Bridges makes Martin Bain claim surrounding Sunderland managerial appointment in 2018
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The Stadium of Light hot seat was vacant following the sacking of Simon Grayson when Sunderland were in the Championship.
Postecoglou is close to being appointed as Neil Lennon's Parkhead successor at Celtic.
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Hide AdBut the 55-year-old Greek-Australian manager was reportedly in consideration by then Sunderland CEO Martin Bain in 2018.
And former Sunderland and Leeds United striker Bridges has revealed that a move was close.
However, he was overlooked by former Rangers’ supremo Bain as he wasn't a big enough name.
Speaking of Postecoglou, Bridges said: "I've lived here [Australlia] for 12 years, I've seen Ange grow as a coach. I've seen what he's done at club level, with the national team, what he's gone and done in Japan.
"He's absolutely incredible.
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Hide Ad"There was an opportunity during the Sunderland Netflix documentary (Sunderland Til I Die), there was a guy there, a CEO called Martin Bain.
"Halfway through the season, I got a phone call off Martin Bain, knowing I was an ex-Sunderland player, knowing that I was living in Australia.
"He wanted a reference on Ange Postecoglou because Ange was in the frame for the job.
"I gave Ange a glowing report. He (Bain) actually said, 'I don't know, I've never heard of this guy'.
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Hide Ad"I said, 'well do your research for one, mate. Have a look at what he's won, have a look at what he's all about. He is perfect for Sunderland football club at this moment in time, he can start you afresh'.
"Their reaction was, 'I need a big name for the fans'.
"That really irked me I said, 'this guy is a big deal, you've just got to make sure that you're doing it for the right reasons'.
"Lo and behold, who got the job? It was the ex-Wales manager Chris Coleman.
"He didn't change anything and I really felt for Ange.
"I think he deserves the opportunity now, I would love somebody to take a punt on him in Europe to let him show what he can do.
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Hide Ad"What that then does is give the credibility back to Australian football for all coaches and all players."