According to the Sun, the Hammers are keen to sign the England midfielder, who was omitted from Gareth Southgate's FIFA World Cup squad - but only for one year. Despite their faith in Wilshere's ability, West Ham are less confident that he can remain injury-free, despite his 36 appearances for Arsenal last season.
Now 26 years old, Wilshere is desperate for regular first time action, having already spent much of his career sidelined with injuries. However, Arsenal's new manager Unai Emery was unable to guarantee the midfielder a regular first team spot, which ultimately precipitated his exit after 17 years with the Gunners - first at the Academy, and later with the senior team, for whom he made 197 appearances and scored 14 goals.
Because he is out of contract, Wilshere will join his next club on a free transfer. In some ways, he would be an ideal signing for the Hammers, who are in need of a creative midfielder in the absence of the injured Manuel Lanzini.
If West Ham's reluctance to sign Wilshere on a long term basis proves to be a deal breaker, Everton would be another possibility - the Toffees' new manager Marco Silva is reportedly interested in the former Gunner.
For all his struggles with injury, Wilshere is still widely admired for his vision and technical ability - and not only in England. His future could potentially lie outside the Premier League, as he has reportedly attracted the interest of Serie A giants Juventus.
The only obvious issue with a move to Turin is that the England star may struggle to break into Juve's first team - and his primary concern is his lack of playing time. Nevertheless, Serie A may prove better suited for Wilshere's cultured style than the Premier League.