Clattenburg sent off two Chelsea players in the 3-2 loss to United at Stamford Bridge and allowed a contentious late winning goal to striker Javier Hernandez, who appeared to be offside when he scored.
"We have lodged a complaint to the Premier League match delegate with regards to inappropriate language used by the referee and directed at two of our players in two separate incidents in today's match," Chelsea spokesman Steve Atkins said. "The match delegate will pass the complaint to the Football Association."
The Professional Game Match Officials Board, which oversees topflight referees in England, said in a statement that it is "aware of the allegations and they are being treated with the utmost seriousness."
"Mark will co-operate fully and welcomes the opportunity for the facts to be established," the statement added. "No further comment will be made until this matter has been properly investigated."
Premier League referees wear microphones in matches and their audio is usually heard by the other match officials during games.
Clattenburg is one of the FIFA candidates to referee at the 2014 World Cup in Brazil.