The former Celtic manager was dismissed by Tottenham Hotspur just over a week after leading them to Europa League success, the club's first major honour in 17 years.
This was not enough to secure his long-term place as manager of Spurs, with the club opting instead for Brentford's Thomas Frank.
According to The Athletic, Brentford made contact with Postecoglou regarding a potential replacement for Frank.
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Informal discussions were reportedly held, but the two parties decided against an official interview, concluding that the move was not in the best interest of either side.
Brentford chose to promote from within, appointing Keith Andrews as their new manager.
The former Republic of Ireland international joined the club last summer as set-piece coach and served under Frank for a year.
Phil Giles, the club's director of football, described the appointment as 'low-risk,' saying [The Athletic]: "It’s more risky to go and get a coach we don’t know from a club where the environment might be different or maybe people around them made them successful.
"This is actually quite a low-risk appointment because I know how good Keith is."