HITC Sport understands that the Hammers will turn to the Croatia international if they fail to land other top targets.
Jelavic is still with Hull City following their relegation from the Premier League and at the moment no deal to take him away from the KC Stadium has materialised.
The 29-year-old has been linked with a number of clubs in Turkey and the Middle East, but now he could be targeted for a move which keeps him in England.
Hull are believed to be looking for just £3million for a player they signed from Everton last year for close to £7million – as reported by the Daily Mirror.
Now HITC Sport understands that Bilic is considering Jelavic, who he handed his international debut to whilst boss of Croatia.
With a Europa League campaign this season, Bilic needs reinforcements and it is the striking department that is looking bare, with injuries sidelining Andy Carroll and Enner Valencia.
And Jelavic is not the only Croatian on Bilic's radar, as HITC Sport understands they have made contact with Barcelona about the availability of Alen Halilovic.