They said they were searching a property in Berkshire, southwest of London, after receiving "an allegation of a sexual nature dating back to the 1980s involving a boy who was under the age of 16 at the time."
Richard, 73, confirmed the property being searched was his and issued a statement denying the allegation.
"For many months I have been aware of allegations against me of historic impropriety which have been circulating online," he said. "The allegations are completely false.
"Up until now I have chosen not to dignify the false allegations with a response, as it would just give them more oxygen," he added.
"However, the police attended my apartment in Berkshire today without notice - except, it would appear, to the press. I am not presently in the UK but it goes without saying that I will cooperate fully should the police wish to speak to me."
Police said there had been no arrests in relation to the search.
Richard, one of Britain's longest-serving entertainers, was born Harry Webb in 1940 and in in his earlier days was often called Britain's Elvis Presley.
He shot to stardom in 1958 with "Move It" and followed up in 1959 with "Living Doll" while cementing his reputation as a heart-throb with starring roles in film musicals "The Young Ones" in 1961 and "Summer Holiday" in 1963.
With his backing group The Shadows, Richard was one of Britain's most successful performers in the pre-Beatles era of the late 1950s and early 1960s.
He has had 14 No. 1 singles in Britain and is the only singer to have topped the UK singles chart in five consecutive decades, from the 50s to the 90s.