The European champions confirmed they have reported Clattenburg to the Premier League match delegate after the fiery 3-2 loss at Stamford Bridge - one part of the allegation concerns language of a racist nature it has been reported by several different sources.
A club 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 [Sunday's] match.
"The match delegate will pass the complaint to the Football Association.
"We will make no further comment at this time."
Clattenburg insisted he would co-operate with any investigation.
Professional Game Match Officials (PGMO), the body which represents Premier League referees, released a statement which read: "PGMO 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.
"No further comment will be made until this matter has been properly investigated."
All four officials - the referee, his two assistants and the fourth official - wear microphones and ear pieces that allow them to hear what each other is saying throughout the match. What is said between them is not recorded.
The revelation that Chelsea had made the complaint rounded off a dramatic day which saw the Blues lose their unbeaten top-flight record after Javier Hernandez struck 15 minutes from time.
Hernandez's goal sparked wild celebrations among the United players which angered a section of the home support in the Matthew Harding stand.
Some Blues fans hurled abuse at the United players who had gathered to celebrate and there were reports some threw objects on to the pitch
A steward was injured during the incident and needed 10 minutes' treatment from paramedics.
Afterwards the club confirmed the steward had been taken to hospital.
"We are looking at reports of objects that may have been thrown or incidents around the steward taking a fall," a Chelsea spokesman said.
"We are investigating all the incidents around that time.
"My information so far is that he slipped and fell, may have hurt his knee, and was being treated by the side of the pitch.
"He's been taken to hospital."