The midfielder came through the Gunners' academy and, barring loan spells at Bolton and Bournemouth, has spent his entire career at the Emirates.
However, he failed to agree personal terms on a new deal and will become a free agent when his contract expires over the weekend.
Wilshere confirmed his departure on social media earlier this month and has been linked with the likes of West Ham, while a switch to Italy has also been mooted.
Reports in Turkey now suggest Istanbul will be his destination following talks with Fenerbahce.
Journalist Volkan Demir says the 26-year-old headed to the banks of the Bosporus earlier this week and, after successful talks, his signing will be announced on Monday.
Wilshere, who has 34 caps for England, is seen as the long-term replacement for former Newcastle midfielder Emre Belozoglu, who is now 37.