Television cameras picked up Wilshere raising his middle finger in the direction of Manchester City fans during the second half of the Gunners' 6-3 defeat at the Etihad Stadium on Saturday.
The FA is reportedly 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 its own disciplinary procedures, the FA 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 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.
Arsenal manager 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.