The star hit out as police carried a five-hour search of his Berkshire penthouse apartment, saying he had seen allegations circulating online for months but had chosen not to “dignify” them with a response.
Sir Cliff, 73, is not currently in the UK but pledged to co-operate with the inquiry, which concerns an alleged assault claimed to have taken place at an appearance by American evangelist Billy Graham in Sheffield in 1985.
It is understood that lawyers for the singer, who is believed to be in Portugal, have been in contact with South Yorkshire Police, who are carrying out the inquiry.
Officers took a number of items from the property for further investigation, the force said.
The investigation is not connected to Scotland Yard’s high-profile sex crime probe Operation Yewtree, which was launched in the wake of the Jimmy Savile abuse scandal.
In a statement yesterday Sir Cliff 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.
“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 co-operate fully should the police wish to speak to me.
“Beyond stating that today’s allegation is completely false it would not be appropriate to say anything further until the police investigation has concluded.”
Speaking at a police station in South Yorkshire, detective superintendent Matt Fenwick stressed the investigation is in its “really early stages”.
He said: “A search warrant has been granted after police received an allegation of a sexual nature dating back to the 1980s. This involved a young boy under the age of 16 years.
“No-one has been arrested at this stage and the owner of the property is not present.”
Conservative MP Nigel Evans, 56, referred to the case in reiterating his call for suspects in sex offence cases to get anonymity and said “questions have got to be answered” over Sir Cliff’s claims the press had been notified about the search.
Mr Evans, who was cleared of a string of alleged sex offences at a trial earlier this year, told Good Morning Britain: “It appears the press knew what was happening before he did and the world’s media were camped outside his doorstep.
“A press helicopter was up before the police even arrived - he is quite right to be angry about that. Questions have got to be answered.
“I believe in the vast majority of cases [suspects] should have anonymity. We have to recognise the impact this has on people.
“People have zero per cent of the facts and 100% of the opinions. It’s quite wrong for people to pre-judge.”
Eight officers from South Yorkshire and Thames Valley police took part in the search of the performer’s flat in a highly-private gated development.
They began work at about 10.30am and five unmarked police cars were seen leaving the site at about 3.30pm.
“A number of items have been removed for further investigation,” a South Yorkshire Police spokesman added.
Born as Harry Webb in Lucknow, India, in 1940, Sir Cliff has become one of the most enduring stars of his time, with hits including Devil Woman and Living Doll.
He was knighted in 1995, the first rock star to be so honoured, and performed at the Queen’s Diamond Jubilee concert at Buckingham Palace in 2012.