Richard, a household name in Britain since the 1950s through hits such as "Living Doll" and "Devil Woman", met officers voluntarily and was interviewed under caution but not arrested or charged.
The interview came after his luxury apartment in Sunningdale, Berkshire, southern England, was last week searched by police over "an allegation of a sexual nature" involving a boy under 16 dating back to the 1980s.
British media have reported that it relates to an alleged incident at a rally by US preacher Billy Graham in Sheffield, northern England, in 1985.
The 73-year-old strongly denies any wrongdoing.
After Richard was interviewed, his spokesman said: "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 probe is not connected to Operation Yewtree, the investigation launched by Scotland Yard into abuse by the late BBC television and radio presenter Jimmy Savile and a string of other ageing celebrities.
But the publicity surrounding the case of Savile, who died in 2011, has led to a surge in allegations of historic sex abuse.
Australian entertainer Rolf Harris, 84, was found guilty in June of indecent assaults against four girls and jailed for five years and nine months for offences between 1968 and 1986.
Richard was born Harry Webb in Lucknow, northern India, in 1940.
He shot to fame in the 1950s with the Shadows and was initially seen as a British version of Elvis Presley.
He has sold more than 250 million records over his career with number one singles in five decades and is nicknamed the Peter Pan of Pop for his youthful looks.
Known for his clean-living, Christian lifestyle, Richard was knighted by Queen Elizabeth II in 1995 and performed at a concert marking 60 years of her reign in 2012.
His fans have rallied around him since the police search, buying copies of one of his singles, "I Still Believe in You", in a bid to get it into this week's Top 40 list of best-sellers.