Richard, 73, whose house was searched earlier this month by police in connection with the allegation, has denied any wrongdoing.
“Cliff Richard voluntarily met with and was interviewed by members of South Yorkshire Police. He was not arrested or charged,” a spokesman for the singer 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.”
South Yorkshire Police confirmed they had spoken to a 73-year-old man “in relation to an allegation of a sexual nature dating back to 1985”.
The man entered the police premises by arrangement, was interviewed under caution but was not arrested, the police said in a statement.
Richard, one of Britain’s longest-serving entertainers, was born Harry Webb in 1940 and in his earlier days was often called Britain’s Elvis Presley.
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.