The former Inter Milan forward is still relatively young at just 21-years-old and obviously has pedigree after playing for Hadjuk Split in his homeland before his move to Italy.
There he went on loan to Lugano and Cesena before a permanent move to Atalanta.
It was during his time with Atalanta that he would score his first and only senior goals in his career so far, notching just four in 31 games.
The good news for Hammers fans is that all of those goals came in Europa League qualifiers, which the club are currently playing in the hope of progressing to the competition proper.
Despite his low goal tally, a big move to Russian side Rubin Kazan on a five-year deal followed last summer.
Big move to Russia has failed
But after failing to register a single goal in 11 games Livaja has now been given permission to train under former Croatia boss Bilic in east London with a view to a move.
It is not yet clear whether any potential deal would be a loan or permanent move to the Premier League side should Livaja impress.
But the Russian giants splashed nearly £4 million on the Croatian less than a year ago so would undoubtedly want some return on their investment.
'Crazy Star' labelled 'overweight and lazy' by boss
The youngster, known as the 'Crazy Star', has struggled to adapt to life in Kazan where most of his 11 appearances came as a substitute.
According to Russian Football News, Rubin Kazan coach Rinal Bilyaletdinov recently criticised the player for being overweight, lazy and having a bad attitude before allowing him to go on trial with the Hammers.
His record internationally for Croatia's youth teams is good, though, as you can see in this video.
He has scored 24 goals in 54 games at youth international level including against an England Under 21 side containing the likes of Harry Kane, Saido Berahino, Luke Shaw and Jack Butland in a Euro Championships play-off last year.
Given Bilic's record of bringing through young Croatian stars and helping them fulfill their promise with the national side, a move to West Ham could be just what Livaja needs to get his career back on track.