Prospect national secretary Alan Leighton said the union did not intend to take legal action against Chelsea, but that the London club should feel morally obliged to make amends via an apology and compensation.
Chelsea are understood to be considering whether there is any action they can take without compromising their position.
Clattenburg will return to match action on Sunday after he was cleared by the FA of the accusation that he said "shut up you monkey" to the Nigerian midfielder John Obi Mikel during Chelsea's 3-2 defeat by Manchester United on October 28.
The 37-year-old will be fourth official for the London derby between Tottenham and West Ham at White Hart Lane and also referee Southampton v Norwich on Wednesday.
Leighton said it was "beyond belief" that Chelsea had pursued the complaint based only on one player - Brazilian midfielder Ramires - saying he heard Clattenburg use the phrase.
Leighton told the Press Association: "Chelsea have done a number of things wrong with this case.
"Firstly, they made the allegation and the nature on it public on the Sunday of the match.
"Secondly, on the Wednesday after their detailed investigation they decided to go ahead with a formal complaint based on one single witness statement, from one person who supposedly heard the offending phrase. The FA statement makes it clear Chelsea didn't even know when the alleged comment was supposed to have been made during the game.
"No organisation in the world would say that was ever going to get past the barrier of evidence needed to bring a charge.
"Thirdly, they have not apologised to Mark, who has gone through hell. This is about an individual who has been badly wronged and there is not even recognition about what they have done to this referee and that is staggering."
Leighton said Prospect did not want to go down the legal route.
He added: "That is not our intention at this moment. Our intention is for something to happen that avoids any need for legal action.
"We have been calling for Chelsea to make an apology and pay compensation because it's the right thing to do."
Sources close to Clattenburg say the referee just wants to put the nightmare of the last few weeks behind him and continue his career with the aim of being selected by FIFA for the World Cup in Brazil in 2014.
Legal action against Chelsea by him or on his behalf by the union would inevitably impact on his refereeing career and could drag on for months if not years.