The Hammers reportedly made a loan bid with a €1 million fee to be paid in January and a release clause worth €8 million.
The Premier League outfit would have been able to sign Yilmaz for a total of €9 million at the end of the season had the deal gone ahead.
Hammers boss Slaven Bilic revealed that Burak Yilmaz was on his transfer list and that he wanted to bring the Turkish international striker to the club.
West Ham eventually ended up signing Fenerbahce striker Emmanuel Emenike on-loan until the end of the season instead.
Chinese side Beijing Guoan were also in talks with Galatasaray over Yilmaz at the same time and reportedly raised their offer following West Ham interest.
The Beijing based side made an €8 million bid upfront to sign Yilmaz and ended up completing the deal last week.
Yilmaz became the second Turkish international to join Beijing Guoan over the winter transfer window with Besiktas defender Ersan Gulum also making the move to China.