The Hammers started the new Premier League season brightly with a shock win at Arsenal last weekend, but a lack of firepower is an issue bubbling away in East London.
With Andy Carroll and Enner Valencia both sidelined through injury, the club are short on back-up for main forward Diafra Sakho, which could be an issue over the course of the next few months.
And reports claim Bilic is eager to address this with a move for Yilmaz.
The 30-year-old has been one of the stars of Turkish football over the course of the past few years, firing Gala to two Super Lig titles (including last term), as well as maintaining an average of better than a goal every other league game for the Istanbul giants since his 2012 arrival.
The current Hammers chief witnessed his talents first hand on multiple occasions while in charge of Besiktas.