Television cameras picked up Wilshere raising his middle finger in the direction of Manchester City fans at the Etihad Stadium during the second half of the Gunners' 6-3 defeat on Saturday lunchtime.
Press Association Sport understands the FA is gathering more evidence to help with the context of the situation having now received the submission of observations from referee Martin Atkinson and his officials.
Under their own disciplinary procedures, the governing body has until 6pm on Tuesday to lodge any formal charge against the player.
Such sanctions have been implemented in the past as was the case with Liverpool striker Luis Suarez, who was suspended for one match, fined £20,000 and warned about 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."
If that is the case, then Wilshere, who could request a personal hearing, will miss the visit of Chelsea on December 23.
Wenger feels the way his men respond from the defeat could well shape the rest of what has so far been a promising campaign.
Indeed Wenger's men may well end up down in fourth spot for Christmas should Chelsea win at the Emirates Stadium - with fifth-placed Everton also out to close up to potentially within just one point if they can win at Swansea.
Wilshere feels Arsenal must learn fast from their disappointing defensive display at the Etihad Stadium, where City made the most of some uncharacteristic poor defending.
"Most of the goals were individual mistakes. I know I certainly am beating myself up about it and the other players I know will be disappointed," Wilshere said on Arsenal Player.
"We will learn from that individually and collectively.
"I think we were too stretched and the space between the defence and the midfield was too much.
"We have to tighten up a little bit."
Arsenal are set to monitor the healing process of the deep cut to defender Laurent Koscielny's knee which was sustained when attempting to prevent City's second goal late in the first half and was unable to continue.