The pair clashed during the 1-1 draw between their sides on Saturday when Mings, 23, landed on the United forward's head with his studs.
Moments later, Ibrahimovic caught Mings in the face with his elbow at a corner.
Both players have until 18:00 GMT on Tuesday to reply to the charge.
"The players were involved in two separate incidents in or around the 44th minute which were not seen by the match officials at the time but caught on video," said an FA statement.
Deliberate elbowing and stamping are both red card offences, so would result in three-match bans if either Ibrahimovic or Mings were found guilty.
However, Mings could face an increased ban for his offence.
The FA statement added: "Furthermore, the FA has submitted a claim that the standard punishment that would otherwise apply for the misconduct committed by the Bournemouth defender is 'clearly insufficient'."
"Off-the-ball incidents which are not seen at the time by the match officials are referred to a panel of three former elite referees.
"Each referee panel member will review the video footage independently of one another to determine whether they consider it a sending-off offence. For retrospective action to be taken, and an FA charge to follow, the decision of the panel must be unanimous."
Ibrahimovic would miss the FA Cup quarter-final against Chelsea and league games against Middlesbrough and West Brom if banned for three games, although he would be available for Europa League matches.
Bournemouth's next three outings are against West Ham, Swansea City and Southampton.
BBC Sports
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