The report claims recently appointed Newcastle boss Steven McClaren thought he was in pole position to land the 25-year-old.
But his old international foe Bilic - who ultimately cost McClaren his England job when his Croatia side famously denied them a place at Euro 2008 - has jumped ahead in the race for the 18-goal striker.
Like Rickie Lambert, Ian Wright and Jamie Vardy, Austin is one of those rare players who made the jump from non-league to the Premier League elite.
In fact as recently as 2009, he was playing for Poole Town where his record of 48 goals in 42 games convinced Swindon Town to take a chance on him.
He has not looked back since, bagging 58 goals in 113 matches for Burnley and QPR before smashing 18 goals in a remarkable debut season in the top flight.
Ironically, Austin's signing could well spell the end of Andy Carroll's injury-plagued time with the Hammers.
Newcastle are said to be interested in a potential deal to bring him back to his boyhood club, who he left for a British record £35 million fee to join Liverpool back in 2011.
West Ham United's most expensive ever player is deemed surplus to requirements by some West Ham fans because in the two years since making his move to Upton Park permanent, Carroll has made just 27 appearances, scoring seven goals.
The likes of the Mirror have reported that the Geordies want to re-sign their former striker.
And that will be music to the ears of some West Ham fans who feel their club simply cannot rely on their injury-prone record £18 million signing anymore.
With every injury - and generally there is a long-term one every season for the 26-year-old - his value rapidly decreases.
So if the Hammers are going to be splashing out in the region of £15 million for Austin, they could be keen to recoup some of that by moving players like Carroll on as Bilic reshapes his squad for a season that has the added demands of Europa League football.