The outcome of last week's disciplinary hearing means the Limerick-born hooker will miss the entire Six Nations championship unless his appeal is successful.
In their written judgement, the Disciplinary Committee found that Flannery had "deliberately stamped on the (Clermont Auvergne ) player's head."
However Munster, while accepting that Flannery did make contact, believe that their player's action was not in any way premeditated or deliberate.
Explaining the decision to appeal, Munster Chief Executive Garrett Fitzgerald said: "After further consideration of the incident, having again spoken to the player, having again studied the video of the incident and taken legal advice, we feel obliged to appeal the findings, of the Disciplinary Committee.
"We do accept that his (Flannery's) boot did make contact but we are fully satisfied that the action was not deliberate."