According to reports in Turkey, the Hammers are in talks with the 29-year-old Trabzonspor captain ahead of the summer transfer window.
Inconsistent performances from Adrian and Darren Randolph have meant that Bilic is on the lookout for a new keeper and Kivrak, who has kept 13 clean sheets in 27 appearances this term, is reportedly their ideal choice.
Kivrak is no stranger to Slaven Bilic - the Hammers boss tried to sign the Turkey international during his spell in charge of Besiktas.
The report also claims that Trabzonspor have already turned down an initial approach from West Ham for their prized shot-stopper.