The Hammers were in the running to sign the Ghana international on a free transfer from Marseille last summer but were eventually beaten to his signature by the Welsh outfit.
The 26-year-old finished his debut Premier League campaign with twelve goals and two assists in 34 appearances but struggled for form towards the end of last season after manager Francesco Guidolin implemented a change in formation.
The prospect of the Irons swooping for the 67-cap attacker was mentioned by The Daily Mail earlier this week, who alleged a potential swap deal involving seemingly exit-bound Enner Valencia.
And The Sun now claim West Ham are set to sign Ayew, with Swansea chairman Huw Jenkins expected to accept a £16million bid from the east London side.
That would constitute a new record transfer fee for the Hammers, slightly beating the £15million they spent on Andy Carroll in summer 2014.