The 17-year-old PSG striker's contract expires at the end of the season.
The report claims that the east London side are willing to sign the PSG forward either in January or in the summer, when his contract expires.
Augustin turns 18 soon and is now free to speak to foreign clubs and the Hammers will reportedly try to sign him 'on the cheap' this month or agree terms for a summer move.
West Ham, however, face competition from a number of clubs, the report states.
Sam Allardyce's side have enjoyed huge success so far this term when it comes to recruitment, with a number of new signings starring for the Upton Park outfit this season.
Allardyce, though, has recently been discussing the element of risk that comes with most signings - and that most certainly applies with youngsters.
Ravel Morrison is the perfect example of a young talent West Ham invested in that has not quite come up trumps just yet.