South Yorkshire Police's Chief Constable has revealed officers are in "regular contact" with Sir Cliff's lawyers over allegations of historical sexual assault.
In a letter to MP Keith Vaz, chairman of the Commons Home Affairs Select Committee, Ch Cons David Crompton said the official investigation into the claims had "increased significantly in size".
South Yorkshire Police has declined to comment on the contents of the letter.
It comes after an independent report criticised an agreement between the force and the BBC which resulted in a raid on the veteran pop star's home being broadcast live on television, saying it "interfered with his privacy" and "may well have caused unnecessary distress".
Sir Cliff has not been arrested or charged with any offence.
The police investigation into veteran singer Sir Cliff Richard has "increased significantly in size" since its inception and involves "more than one allegation", South Yorkshire's Chief Constable has revealed.
David Crompton said the "expanding nature" of the investigation meant he could not give a date when it would be concluded. In a letter to Keith Vaz, chairman of the Commons Home Affairs Select Committee, Mr Crompton said the force was in regular contact with Sir Cliff's lawyers.
Rotherham Council's interim chief executive says the authority accepts the damning criticism it faced in yesterday's inspection report.
Jan Ormondroyd said the report contained a "catalogue of cultural and system failings", and apologised for the "devastating impact that this has had on the lives of the people of Rotherham".
She added that the council needed a "move towards stability" and "the fresh start which [report author] Louise Casey and her team have called for".
We understand the impact of the current uncertainties around the future leadership and management of the Council on the people of Rotherham, and also on staff in the Council. I have written to the Department for Communities and Local Government, urging them to confirm urgently who the five Commissioners who will take over the running of the Council will be and to ensure they are in place as soon as possible.
The mother of a girl who was abused as a teenager in Rotherham has told ITV News the outgoing council failed to do their jobs and protect the town's children.
After the council was declared "not fit for purpose" in a damning report, the mother said: "I wonder how they would feel if it was their daughter - the police and the council want to think about how many lives they have ruined.
"They just would not listen, and they did not do their jobs and protect our children."
ITV News Social Affairs Editor Penny Marshall reports:
A Rotherham councillor has admitted the council did not act "quickly enough" to tackle child sexual exploitation in the South Yorkshire town.
Since the publication of the Jay Report, the council has begun to take measures to improve the support for victims. bringing in new management, and to tighten licencing controls.
Sadly, as the report today makes clear, we have not been able to do enough, quickly enough.
When I spoke to one of the Rotherham abuse victims this evening I asked whether she felt any safer now after today's developments.
She replied: "Safer, but not safe enough".
Six Whitehall-appointed commissioners will come to Rotherham to effectively run the council.
Westminster expect there to be local elections in 2016 when the people of this town will be given a chance to elect people they trust.
But this affair is far from over - the National Crime Agency has said today there may be grounds for criminal investigations, possibly against a police officer and two councillors.
Rotherham Council says it needs time to "understand and respond" to today's damning report, as well as the "implications" of its cabinet's resignation and the intervention package proposed by Eric Pickles.
In a statement, a spokesman for the council said: “We recognise the need for a fresh start that is so clearly identified in the report, but also appreciate that we need to continue to deliver services to the people of Rotherham, and ensure business continuity.
“We should not forget that the publication of this report will re-open old wounds for the victims and survivors of CSE. We will continue to put in place the help and support they need at this difficult time, including our dedicated helpline.”
Paul Lakin, leader of Rotherham Borough Council has resigned with immediate effect.
Cllr Lakin - who only took on the top role last year - will also stand down as local councillor for Rotherham's Valley Ward, the council said in a statement.
The rest of the cabinet are due to resign their positions "as soon as transitional arrangements are put in place", according to an earelier statement.
Louise Casey, the head of the Government's Troubled Families Unit, headed the inspection into Rotherham Borough Council and says she was shocked by what she found.
South Yorkshire's former Police and Crime Commissioner Shaun Wright refused to be interviewed as part of an inspection of the borough's council, the author of today's damning inspection report said.
Wright - who had been a councillor in the town and responsible for children's services - resigned in the wake of a previous report by Professor Alexis Jay, which claimed that more than 1,400 children had been abused in the area between 1997 and 2013.
The council's former leader Roger Stone also declined an invitation for interview.
Despite other resignations in the wake of that report - such as that of chief executive Martin Kimber and director of children's services, Joyce Thaker - Louise Casey's more recent inspection found a continued denial of any wrongdoing among current council leaders.
When inspectors commenced work in Rotherham, we were struck by the overwhelming denial of what Professor Jay set out in her report. This attitude was so prevalent that we had to go back through many of the aspects of her work in order to satisfy ourselves that the council had no grounds upon which further action could be delayed.
Casey added: "When asked, 70% of the current Rotherham councillors we spoke to (including those in the Cabinet) disputed Professor Jay's findings."