The referees union Prospect has called for Chelsea to compensate Clattenburg, after he was cleared of the accusation that he said "shut up you monkey" to the Nigerian midfielder during Chelsea's 3-2 defeat by Manchester United on October 28.
It is understood Chelsea will resist such calls - and the club have pointed out that the FA have agreed their Brazilian midfielder Ramires, the sole source of the complaint, was acting "in good faith".
Prospect's national secretary Alan Leighton said: "Mark Clattenburg deserves a full and unreserved apology from Chelsea. He deserves compensation for loss of earnings, damage to his reputation and stress.
"In addition, Chelsea should make a sizeable donation to Show Racism the Red Card and Kick It Out to help the fight against racism in football and beyond.
"This was a charge brought on the flimsiest of evidence that should never have got to this stage."
Clattenburg himself spoke of the "truly frightening" fear that his career could have been wrecked by an unjust accusation.
The match official also implicitly criticised Chelsea for going public with the complaint before the FA had even launched a formal investigation.
Clattenburg said: "To know you were innocent of something but that there was the opportunity for it to wreck your career was truly frightening.
"Racism has no place in football and this experience should not discourage those to speak out if they genuinely believe they are a victim of abuse.
"However, there are processes that should be adhered to in order that any investigation can be carried out in a manner that is fair for all parties involved.
"I know first-hand the ramifications of allegations of this nature being placed into the public domain ahead of a formal process and investigation. I hope no referee has to go through this in the future.
"I am looking forward to putting this behind me and concentrating on refereeing in the Premier League and other competitions."
Mikel has himself been charged by the FA for using threatening words or behaviour - an alleged breach of FA Rule E3 - after going into the referees' changing room after the match in question.
Chelsea insisted that they had had a duty to report the claim against Clattenburg, but that Mikel would not deny the charge.
A club statement said: "The FA states Chelsea took the correct action following the Manchester United match and encourages all players who believe they have been either subject, or witness, to discriminatory abuse to report the matter immediately to the match officials on the day, and in turn to the FA.
"The club also notes the charge brought against John Mikel Obi. While the player does not deny the charge, he will request a personal hearing to explain the mitigating circumstances.
"With regards to the Mark Clattenburg decision, the club accepts the case is now concluded and notes the FA states the allegation was made in good faith."