The west London club lodged an official complaint to the FA following their 3-2 defeat to Manchester United in October claiming Clattenburg used "inappropriate language" towards midfielder John Obi Mikel.
However, neither the club nor player forwarded their allegations onto the police who have subsequently dropped their investigation into the Premier League official, citing a lack of evidence as a primary reason.
And Herbert has condemned the club's approach and believes it is further proof of football's inability to effectively deal with racism.
He told Sky Sports: "What seems to have happened is that although Chelsea made a report to the FA they've decided, in their infinite wisdom, not to submit the same papers to police.
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"If there is anything we can do to put pressure on the victims concerned to go forward to the police with this matter, then we'll do that.
"It highlights the fact that football cannot regulate itself and really, on this issue of race, always wants to take the path of least resistance.
"Given the publicity this engendered it's fair for all concerned, including the person who's alleged to have made the comment, that justice is seen to be done and it's not dealt with by the FA internally."