Former Galatasaray sporting director Bülent Tulun told the Turkish sports newspaper that Hutchinson has all but completed his move to the East London based club.
Besiktas are however, yet to confirm the deal and the experienced midfielder is currently in Canada.
The Hammers have been closely linked with Besiktas players over the past few weeks.
West Ham boss Slaven Bilic is no stranger to the Istanbul giants having coached the side for two-seasons prior to his move to the Premier League.
The Hammers are already reported to have reached an agreement with Besiktas over Gokhan Tore and are also interested in star midfielder Oguzhan Ozyakup.
Atiba has one-year remaining on his current deal but has has to extend his current deal and there have been rumours of a summer exit.
The 33-year-old has however, been a key player for Besiktas and played a vital role in the side that won the league title last season.