The Ghanian joined the Welsh club last summer and was a star performer, scoring 12 league goals, including on his debut at Chelsea. He finished the year as Swansea’s top league scorer for the season, 1 ahead of Gylfi Sigurdsson.
It was the second consecutive season Ayew netted in double figures, after scoring 10 Ligue 1 goals for Marseille before making the transfer to the Liberty Stadium.
But the 26-year-old is keen for the Swans to accept the £20million offer from West Ham for his services, so he can make the switch to the London club, who finished 7th last season.
The Hammers have been desperate to add a forward this summer, but have struggled, with their long-term pursuit of Carlos Bacca still failing to lead to any joy.
The club were close to selling Diafra Sakho to West Brom, only for the deal to fall through, with the BBC reporting a bank injury was the reason the £15million deal was called off.