Television cameras picked up Wilshere raising his middle finger in the direction of home fans at the Etihad Stadium during the second half of Saturday's 6-3 defeat for the Barclays Premier League leaders.
Following the submission of observations from Atkinson and his officials, the FA will then consider if the matter requires separate action to be taken.
Under the FA's disciplinary produces, sanctions can be implemented as was the case with Liverpool striker Luis Suarez, who was suspended for one match, fined £20,000 and warned as to his future conduct after he made a gesture to home fans as he walked off the pitch following a 1-0 defeat at Fulham in December 2011.
Gunners boss Arsene Wenger admitted in his post-match press conference that "if he did it and he is banned, then we will have to accept it."
Arsenal endured a testing afternoon at City, where they had no answer to the Blues' formidable frontline which saw Sergio Aguero fire the hosts ahead on 14 minutes.
The hosts had a number of chances to increase their lead, but Arsenal did respond to equalise with the first of two goals from the returning Theo Walcott, starting his first game since September after abdominal surgery.
However, Alvaro Negredo and Fernandinho added more for City and although Walcott's fine second kept the visitors in contention, David Silva and Fernandinho put the result beyond doubt before Per Mertesacker grabbed a consolation and Yaya Toure completed the scoring from the penalty spot.
Germany international Mertesacker, who captained the side, also remonstrated with countryman Mesut Ozil for not applauding visiting fans at full time.
The Arsenal club record signing issued an apology on his Facebook page on his website, www.mesutoezil.com: "Sorry I didn't thank the fans at the end of the game!
"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."
The defeat at City saw the Gunners' lead at the top of the 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."
Wenger remained determined to put on a brave face on a potentially damaging reverse on the belief of his side, whose character to last the distance will now again be called into question.
"They scored the goals, you cannot say they haven't scored them, but we have to look at ourselves," Wenger said.
"Defence starts up front and that is where we were very poor, and in midfield as well. The defenders were exposed too much.
"It takes teamwork to defend. Until now we have respected it is not about you, it is about the team. Collectively we were poor."
Arsenal were also waiting on updates on the injury sustained by defender Laurent Koscielny, who suffered a deep cut attempting to prevent Negredo scoring City's second late in the first half and was unable to continue.