Officers raided the home of Richard, 74, in Berkshire, west of London, last August, when he was on holiday, over allegations which dated back to the 1980s and involved an under-age boy.
Richard was later interviewed under caution but not arrested or charged with any crime. He said on Wednesday all the allegations were "absurd and untrue".
David Crompton, the Chief Constable of South Yorkshire Police, said his force's inquiry had now expanded.
"This is an investigation which has increased significantly in size since its inception," he wrote in a letter to the chairman of parliament's Home Affairs Committee which was written two weeks ago but only made public on Wednesday.
"Sir Cliff Richard's lawyers are aware that there is more than one allegation," the letter added. "It would be premature and potentially misleading to predict a likely date when it will be concluded. However, we are progressing as swiftly as possible."
Richard, born Harry Webb in 1940 and in his earlier career referred to as Britain's Elvis Presley, 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.
He shot to stardom in 1958 with "Move It" and cemented 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.
"I have no idea where these absurd and untrue allegations come from," Richard said in a statement. "I have never, in my life, assaulted anyone and I remain confident that the truth will prevail."
The raid on his house last year was filmed by BBC cameras after the broadcaster was given advance warning, leading to criticism from lawmakers who described that co-operation as "inept" and causing "irreparable damage" to the singer's reputation.