The 33-year-old spent five years on the books at Manchester City, but was on loan at Everton last season and flourished in the Toffee’s midfield.
He was a mainstay in the side, making 32 appearances in the Premier League under Roberto Martinez, scoring three and assisting four goals in a hugely successful season that saw Everton finish in fifth place amassing a club record 72 points.
Barry is now a free agent and is the subject of offers from Everton, Arsenal and now West Ham as he looks set to remain in the Premier League.
The Daily Mail are reporting that West Ham have offered Barry around £65,000 a week in a bid to convince him to make the move to East London, significantly more than their rivals in the chase for the former Aston Villa man.
Barry would play regular first team football under Sam Allardyce, something which could influence his decision as he would not see enough game time at the Emirates, while Everton are said to be looking at other options.
West Ham’s midfield would be strengthened by the arrival of Barry, and the midfielder is showing no signs of slowing with age as he completed more passes than any other Everton player last season.
Diame has been linked with a move away, and is statically inferior to both Noble and Barry, who were substantially better in the middle of the park.
Diame made less interceptions and chances, as well as recording a lower number of tackles and headers won, compared to the English pair.
Barry and Noble could form a solid partnership that could improve the Hammers stability, and would allow a more creative player to play ahead of them.
West Ham should cash in on Diame if they can secure the services of Barry this summer, and it would have to be seen as a shrewd piece of business if the case.