The east London based side are reported to have offered a three-year deal worth €2.5 million per-season and a €3 million sign-on fee.
The Turkish sports newspaper claims that Hammers boss Slaven Bilic has been keen on Gonul since his time managing Besiktas and that he is on high alert due to the Turkish international being into the last six-months of his contract.
West Ham can currently sign Gonul on a free-transfer but Fenerbahce are reported to be keen on trying to extend his current deal.
Gonul has consistently been one of the Yellow Canaries highest performing players over the last eight years winning two league and two Turkish Cup trophies during his time at the club.
The 31-year-old has a wealth of international and European competition experience having reached the quarter-finals of the 2007-08 Champions League and the Europa League semi-finals in 2013.
Gonul has one goals and four assists in all competitions for the Istanbul giants this season and is almost certain to join Turkey in the European Championships this summer.