Gibbs was dismissed in the first half of the 6-0 defeat against Chelsea at Stamford Bridge for handball, despite the fact it was team-mate Alex Oxlade-Chamberlain who dived to push an Eden Hazard shot past the post.
Referee Andre Marriner, with the apparent aid of his assistant official, made the call to show Gibbs the red card despite protests from Oxlade-Chamberlain and the Arsenal players.
The FA confirmed that there was provision for Arsenal to appeal.
"Clubs can appeal mistaken identity to The FA, although The FA can pro-actively review also," an FA tweet read.
"Following a red card clubs can also review possibility of a wrongful dismissal claim to The FA, unless dismissal is for 2 yellow cards."
There is a precedent for the decision to be changed after League 1 side Preston had the wrong man sent off against Port Vale in November.
Striker Joe Garner was suspended retrospectively for three games after midfielder Neil Kilkenny had mistakenly been sent off.
Gibbs' mistaken red card would not have happened under UEFA's extra-officials policy, according to a spokesman for president Michel Platini.
It is a situation that Pedro Pinto, the chief of press for Platini, said could have been avoided if there was an extra assistant on the by-line.
Pinto wrote on Twitter: "What confusion at Chelsea-Arsenal with the pen decision. That's why over 30 European countries have endorsed UEFA-backed 5 officials system.
"With an additional assistant referee on the end line, referee would not have got that sending off wrong. Technology is not the answer..."
Pinto claimed that "more eyeballs" were the key to improving referee decision-making rather than technology alone.
The Premier League has employed goal-line technology for the first time this season - which would not have aided Marriner's decision at Chelsea because it determines whether a shot has crossed the line only.
Pinto added on Twitter: "More eyeballs are the answer. GLT helps with goal line decisions, but five officials system gives referee more angles of vision. It's not one vs the other."