With Enner Valencia and Andy Carroll both sidelined through injury, manager Slaven Bilic has been left with a dearth of attacking talent at his disposal, prompting a search for a new centre forward.
Now, it seems that Yilmaz is the top target. The Turkish international has smashed 73 goals in 121 games for Galatasaray since joining from Trabzonspor in 2012, and will be familiar to Bilic given that the Hammers boss managed Besiktas - one of Galatasaray's rivals in Turkey.
Despite that rejected bid, West Ham seemingly aren't ready to give up the fight for Yilmaz just yet, with NTV Spor reporting that the Hammers are poised to lodge an improved bid.
The report states that West Ham will up their bid from €5m to €7m (£5m) in a bid to tempt Galatasaray into a sale, despite the 30-year-old's importance to their side.
Yilmaz is a well built striker with superb predatory instincts inside the box, and with 18 goals in 40 games for the Turkish national team, he has not only proven himself at club level but also on the international stage - making him a very appealing target for the Hammers as they search for a new striker.