The Manchester City defender posted a picture of Manchester City players celebrating a goal in the win over the Clarets, with the comment: "10 against 12... but still fighting and winning as a team".
The post appeared to be a swipe at referee Lee Mason, who sent off Fernandinho for Man City, meaning that the Citizens were made to play with 10 men, while he also appears to suggest that Burnley had an extra man with the help of the referee.
FA rule E3 states: "Comments about match officials which imply bias, attack the officials' integrity can be considered in breach of the rule".
This means that Sagna is likely to face punishment for his post on Instagram, with a fine and a warning the most likely sanction.
The Frenchman has reportedly apologised for the post.