That followed talks with new Gunners boss Unai Emery who reportedly told Wilshere his game time would be "significantly reduced" if he chose to remain at the Emirates.
West Ham are desperate for quality reinforcements this summer after a torrid couple of seasons flirting with relegation since the big move to the London Stadium.
Despite his injury problems many Hammers fans were excited at the prospect of signing Wilshere, who is a boyhood West Ham fan.
But according to a new report from ESPN, Fenerbahce have jumped to the front of the queue and are in 'advanced talks' with the player.
According to the report, Fenerbahce's sporting director Damien Comolli is leading the negotiations and is prepared to offer Wilshere a deal worth €4million a season.
There was some hope for the Hammers from the report, though, which claimed Fener were having to work on convincing Wilshere to make the switch, including offering an attractive sign-on fee, but have a few concerns, namely the fact Wilshere has never played abroad and would have to uproot his family to move to Istanbul.