And Prospect have also decided to drop their demand for Chelsea to pay compensation to the official.
After a match against Manchester United on 28 October, Chelsea complained that Clattenburg had used inappropriate language towards midfielder John Obi Mikel.
The claims prompted huge media interest but Clattenburg was fully cleared by the FA last week, who said he had no case to answer.
And Chelsea have now offered their regret at the statement they released after the match and at putting Clattenburg and his family under such intense media scrutiny.
And the refs' union have now issued their own release, which read: "Prospect National Secretary Alan Leighton said the use of the word "regrets" in relation to the swift issue of the club's statement publicising the allegation and in relation to the impact on Mark and his family, were tantamount to and accepted by Mark as an apology.
"Importantly the statement also refers to Chelsea's willingness to welcome Mark back to Stamford Bridge.
"Prospect had called for Chelsea to pay compensation to Mark but has now dropped that demand on the basis that Mark has received the response that he sought. Mark now looks forward to returning to his love of refereeing."
Leighton added: "This was an important move forward in confirming Chelsea's recognition of Mark's integrity and impartiality."
The Chelsea statement was released after a meeting between their Chairman Bruce Buck, Premier League Chairman Richard Scudamore, Mike Riley of Professional Game Match Officials Ltd, and the 16 Select Group Referees.
And Leighton said the referees "welcomed the opportunity to express their concerns about the way that Mark had been treated.
"In a thoroughly professional manner they explained what the impact had been in real terms and that there could be no repetition of the events."