73-year-old Richard, one of Britain's most successful entertainers, met police officers by appointment after police searched his home in Sunningdale, Berkshire, on 14 August as part of their investigation involving a boy under 16 dating back to the 1980s.
The allegation against Richard relates to an assault claim at an event featuring US preacher Billy Graham at the Bramall Lane stadium in Sheffield in 1985, the BBC reported.
The veteran pop star, well-known globally since the 1950s through hits such as "Living Doll" and "Devil Woman", denies the alleged offence, saying the claim of an assault at a religious event in Sheffield in 1985 is "completely false".
A spokesman for Richard said: "Today Sir Cliff Richard voluntarily met with and was interviewed by members of South Yorkshire Police. He was not arrested or charged.
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"He co-operated fully with officers and answered the questions put to him.
"Other than restating that this allegation is completely false and that he will continue to co-operate fully with the police, it would not be appropriate for Sir Cliff to say anything further at this time."
The investigation is not connected to Operation Yewtree, the probe launched by Scotland Yard into abuse by the late BBC television and radio presenter Jimmy Savile and a number of other ageing British celebrities, media reports said.
Considered a life-long bachelor, Richard has represented the UK in the Eurovision song contest twice and in 2013 released the 100th album of his career.
He was knighted in 1995 and performed at the Queen's Diamond Jubilee concert at Buckingham Palace in 2012.
His worldwide hits include songs like 'Congratulations' and 'Summer Holiday'.