The England international has dropped down the pecking order at the Emirates following numerous injury woes over the years.
Last season, the 25-year-old joined Bournemouth on loan, but the stint ended in disappointment when he fractured a bone in his leg.
Since recovering, Wilshere has made just one appearance for the Gunners, and that was a cameo in Thursday's 3-1 win over FC Koln in the Europa League.
According to the Daily Star, after failing to make the squad for Sunday's goalless draw with Chelsea, Wilshere has accepted that his future lies elsewhere.
The report claims that the midfielder is keen to impress West Ham in the hope of convincing the East London outfit to make a move for him in January.
Wilshere is currently in the final year of his contract at Arsenal.