The Hammers have reportedly been tracking Gunok, who has five caps for Turkey, for some time and they feel he would make a good addition at the Boleyn Ground.
Although Besiktas and Manchester United are also said to be interested, the reports claims that West Ham believe they may be able to convince the goalkeeper into a move to London.
The 26-year-old is expected to be available on a free transfer this summer and he could make a bargain buy for the Hammers.
Impact for West Ham?
It seems unlikely that Gunok would displace Adrian as West Ham's first-choice in goal, but he could represent a decent back-up option.
At the moment Jussi Jaaskelainen is the Hammers' second choice in net, but he is reaching the end of his career and may need to be upgraded for the future.
It means that Gunok could represent a cheap, low-risk signing for West Ham and he could solve a potential problem, without costing them much of their budget.