The report claims that the Hammers have sent representatives to Turkey to complete the deal before the transfer deadline.
The Premier League club are in the transfer market for a striker as forwards Enner Valencia, Mauro Karate and Andy Carroll are all currently out injured.
The Metro recently reported that West Ham United had agreed personal terms with the Turkish international.
The report claims that the prolific striker has agreed to a £6.4 million per season deal with the London based side.
However, Galatasaray turned down the bid and stated that they would be keeping hold of the prolific striker.
The Lions would be unable to replace Yilmaz as the Turkish transfer window has come to an end.
Galatasaray sold Felipe Melo and Alex Telles to Inter Milan on transfer deadline day which ended on 30 August 2015 in Turkey this season.