The admission follows a meeting which took place on Monday afternoon at St George's Park between Professional Game Match Officials Limited (PGMOL), the Premier League and Chelsea chairman Bruce Buck.
The FA dismissed complaints last week that Clattenburg had abused Jonn Obi Mikel during Chelsea's defeat to Manchester United in October.
In return the referees say they understand the club was duty-bound to make a complaint given the allegations.
A joint statement made by all parties said: "There was a constructive and open discussion. The club regrets not having given more consideration before issuing a statement on the evening of Sunday 28th October.
"The club also regrets the subsequent impact the intense media scrutiny had on Mark Clattenburg and his family.
"The referees accept that, given Chelsea FC had received a good faith claim from one of their employees, the club had an obligation under FA rules to report the allegation.
"There was recognition by all parties that the impartiality and integrity of refereeing in this country remains paramount.
"Chelsea FC made it clear they would welcome Mark Clattenburg back to Stamford Bridge in the future and PGMOL would have no issue in appointing him to a Chelsea FC match going forward.
"It was a thoroughly professional meeting. All parties now believe it is time to draw a line under this incident, learn from it and move on for the good of all Premier League clubs, players and match officials."