The 33-year-old striker has been a free agent since leaving Newcastle United at the end of last season, concluding a five-year spell at St James' Park.
West Ham have emerged as the frontrunners for Wilson's signature, ahead of competition from Leeds United and several clubs in Saudi Arabia.
According to reports, Wilson turned down a pay-as-you-play style contract from Newcastle in favour of testing the free agent market, a decision that now appears to be paying off.
The move to the London Stadium was facilitated by Wilson's agent, Will Salthouse, who maintains a close relationship with West Ham co-owner David Sullivan.
Salthouse is said to have directly approached Sullivan about the possibility of bringing the England international to east London.
West Ham's offer includes a minimal base salary, with performance-based bonuses for goals and appearances. The financial package reflects both Wilson's experience and the club's strategic approach to managing costs while strengthening their attacking options.
If finalised, Wilson will become the Hammers' third addition of the summer, following the arrivals of midfielder El Hadji Malick Diouf and full-back Kyle Walker-Peters.
The veteran forward is expected to provide depth and competition up front as West Ham prepare for the new Premier League campaign.