Along with Alexis Sanchez and Mesut Ozil, Wilshere's current contract expires at the end of the season, meaning he can leave on a free transfer next summer.
Club officials know they need to tie the midfielder down to new terms as soon as possible, or they risk another example of poor planning for the future.
Wilshere is returning to his best form with Arsenal and could soon be in line for a sensational return to Gareth Southgate's England side, who are struggling for creativity in the middle of the park.
The 25-year-old came on as a late substitute and provided an assist for Aaron Ramsey in Arsenal's recent win over Everton, and is likely to start against Norwich in the League Cup next week, which will be another opportunity for him to prove his fitness.
With Premier League rivals Chelsea and Manchester City likely to reignite their interest in Wilshere, Arsenal know they need to make substantive progress in negotiations by the end of next month.