The 26-year-old, who is out of contract this summer, has been heavily linked with an exit from the Emirates Stadium after contract negotiations stalled, with a flurry of Premier League sides said to be interested.
Wilshere recently took the Gunners name out of his Twitter bio, fuelling rumours of an imminent exit from the north London side. He subsequently followed the official Twitter account of West Ham, as well as club captain Mark Noble and former Hammers striker Dean Ashton.
Wilshere was recently one of the most high-profile omissions from England's World Cup squad, despite making 38 appearances in all competitions for Arsenal this season.
The midfielder has previously struggled with a string of injury problems, which have hindered both his domestic and international career.
West Ham are yet to make their first signing since the appointment of Manuel Pellegrini as manager and could look to bring in Wilshere for free as the Hammers look to improve on their 13th place finish in last season's Premier League.
Wilshere's social activity could mean nothing but it certainly has the feel of a more pointed statement by the midfielder, who has a history of being outspoken on social media.
Arsenal, meanwhile, have begun their summer transfer dealings by signing veteran defender Stephan Lichtsteiner from Juventus on a free transfer. The 34-year-old Swiss international becomes the first signing of new manager Unai Emery as the club looks to move on from the reign of Arsene Wenger.