New developments in the case were revealed in a letter from South Yorkshire Police to Home Affairs Committee Chairman Keith Vaz.
The letter, dated 10 February, says: "South Yorkshire Police detectives are in very regular contact with Sir Cliff Richard's lawyers. Typically this involves a verbal update about once a fortnight.
"This is an investigation which has increased significantly in size since its inception. Sir Cliff Richard's lawyers are aware there is more than one allegation.
"In view of the expanding nature of the investigation, it would be premature and potentially misleading to predict a likely date when it will be concluded."
A statement released by Sir Cliff said: "I have no idea where these absurd and untrue allegations come from.
"The police have not disclosed details to me. I have never, in my life, assaulted anyone and I remain confident that the truth will prevail. I have co-operated fully with the police, and will, of course, continue to do so.
"Beyond stating that the allegations are completely false, it would not be appropriate for me to say anything further until the investigation has concluded, which I hope will be very soon.
"In the meantime, I would, again, like to thank everyone for supporting me through this unbelievably difficult period."
Sky News understands the accusation against the performer, which he was questioned about last year, dates back to an event held by American evangelist preacher Billy Graham in Sheffield in 1985.
A report by the Home Affairs Select Committee in October said a highly publicised raid on the performer's home in August was "inept" and caused "irreparable damage" to his reputation.