The 33-year-old midfielder spent last season on loan with the Toffees, but despite his impressive displays Manchester City have opted against extending his stay at the Etihad Stadium.
Everton have been linked with a deal to keep him on a permanent basis, while Arsenal are also thought to have made him an offer after being wowed by his evergreen performances.
Now West Ham are also in the battle, with Sam Allardyce looking to add more creativity to his ranks.
The Hammers chief has been told that he must promote a more engaging style of play at Upton Park after fans spoke out against his direct philosophy, and Barry is seen as a good central option.
Everton and Arsenal are thought to be willing to pay the former Aston Villa man around £50,000-per-week, but the Irons are prepared to go £15,000-per-week higher and guarantee him first-team action to push through a deal.
West Ham may also move for another Manchester City freebie in the shape of Joleon Lescott.