Two accusations of using "inappropriate language" have been levelled against Clattenburg, who sent off Branislav Ivanovic and Fernando Torres during United's stormy 3-2 win.
Chelsea did not go into details, but a spokesman confirmed two separate incidents involving two different players have been reported.
And Sky Sports News' chief reporter Bryan Swanson identified one of the two players involved as midfielder John Obi Mikel, who was seen attempting to enter the referee's dressing room after the game.
The spokesman said: "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.
"The match delegate will pass the complaint to the Football Association. We will make no further comment at this time."
In a later statement the Professional Game Match Officials - the body which controls Premier League referees - said it was "aware of the allegations and they are being treated with the utmost seriousness.
"Mark will cooperate fully and welcomes the opportunity for the facts to be established".
Clattenburg was wearing a microphone during the game to help him communicate with his assistants, but conversations during the game are not recorded.