Wilshere's current deal at the Emirates Stadium expires this summer, but talks over an extension have so far proved unsuccessful and the 26-year-old will officially be a free agent next month.
The England international, who has been left out of the squad for this summer's World Cup, was initially unwilling to take a pay cut, but Arsenal are understood to have since returned with an improved offer.
The Mirror reports that Wilshere is now willing to put pen to paper on the deal which has been offered to him, which is expected to extend his stay in North London to 2021, with the option of a further 12 months on top of that.
The player had previously suggested that the identity of Arsenal's next manager could have an impact on his decision, and the Gunners are understood to be close to appointing Mikel Arteta as Arsene Wenger's successor.
Wilshere has been at Arsenal since the age of nine, making almost 200 first-team appearances for the club.