The Football Association will review the full match report of referee Martin Atkinson before deciding whether to launch a disciplinary case against Wilshere after television cameras showed him raising his middle finger in the direction of home fans during the 6-3 loss against Manchester City at the Etihad Stadium.
Arsenal manager Arsene Wenger says the club will accept any punishment the FA hands down.
Liverpool striker Luis Suarez was suspended for one match, fined £20,000 and warned as to his future conduct after he made the same gesture to home fans as he walked off the pitch following a 1-0 defeat at Fulham in December 2011.
A possible ban for Wilshere could be compounded by bad news on the injury front, with Arsenal awaiting an update on Laurent Koscielny after the defender suffered a deep cut attempting to prevent Alvaro Negredo scoring City's second goal.
The heavy defeat at City saw the Gunners' lead at the top of the Barclays Premier League cut to two points, with the visit of second-placed Chelsea next up on December 23.
Wenger believes his side's response to the bitter disappointment of Saturday's humiliation could define their season.
"It is how we respond in the next game, that is what makes a season. It can happen," Wenger said.
"You could see we played on Wednesday away from home and we were not as fresh as we can be, but it is how we respond now.
"There are always positives in a game and there were a lot of them, but still we have to come back to reality.
"It is about us defending together in games like that and we did not do that."
Meanwhile, midfielder Mesut Ozil has apologise for failing to applaud the visiting fans at full-time of Saturday's match.
"Sorry I didn't thank the fans at the end of the game!" Arsenal's record signing said on his Facebook page.
"You have been brilliant to me and I know you had travelled a long way and spent your money to support us.
"I was upset with the result and know I should have come to you to say 'thank you' and I know it is a big Arsenal tradition win, lose or draw."