The Hammers have made no secret of the fact they are desperate to sign two strikers this summer but so far only ageing defender Pablo Zabaleta has arrived on a free from Manchester City.
Bilic has been strongly linked with moves for a number of strikers including Kelechi Iheanacho, Olivier Giroud, Michy Batshuayi, Carlos Bacca, Anthony Modeste and Cedric Bakambu.
The club appears to be making little headway in those deals, though, prompting talk of a move for back-up target Andre Gray.
But could Bilic use Enner Valencia's impending move back to Mexico with Tigres to demand a swap deal for a striker he wanted to sign the last two years?
Enner Valencia is on his way out of West Ham
According to Sky Sports West Ham have agreed a fee of between £6-7m with Tigres for Valencia - relative small fry nowadays considering the Ecuadorian star is still highly rated at international level.
And yesterday French media outlet L'Equipe claimed that LA Galaxy had failed with a bid to bring France and Tigres forward Andre-Pierre Gignac to MLS, as reported by Get French Football News.
So should Bilic demand the big forward, who has scored 53 goals in 81 appearances in Mexico since leaving Marseille, be included in any deal.
While the 31-year-old would probably not be first choice when the likes of Andy Carroll are fit, he knows where the goal is and has a superb injury record, something which will count in his favour in East London.
There may be one sticking point, though, Gignac famously fell out with former Marseille teammate Andre Ayew who is currently at West Ham, as reported by the Daily Star.
That was two years ago, though, and a lot of water has passed under the bridge since so old wounds could well be healed by now.
And West Ham's desperation may leave them with little choice.