In a video released by the Falsely Accused Individuals for Reform (FAIR), who advocate for justice and law reform for those falsely accused of sexual offences, Sir Cliff said: "It's not just for celebrities, it'll be for the everyday man in the street."
He added: "I'd had this false accusation and I can only tell you what it would have done for me had I not been named.
"I was never charged, I was never arrested. It still bugs me that other people, men and women are going to find themselves in that same position.
"The only way to help them is to make a change in the law, the rules aren't correct."
Cliff called for a change in the law on sex-offence investigations, after accusations of child sexual assault made against the singer, were broadcast by the BBC.
The proposed Anonymity (Arrested Persons) Bill would make it illegal for those arrested for alleged sexual abuse to be publicly named until charges have been brought.
The parliamentary petition supported by FAIR, has been signed by over 24,000 people since it went live 13 days ago.
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