The home secretary, Theresa May, is to announce a wide-ranging inquiry into public bodies' and institutions' handling of historical child abuse allegations, with David Cameron saying there will be no stone left unturned in the search for the truth.

Downing Street drew parallels with the panel inquiry into the 1989 Hillsborough stadium disaster – and its subsequent handling by public bodies including the police – rather than the judge-led Leveson inquiry into the practices of the press.

The prime minister's spokesman said there was a case for examining how a range of public bodies have handled child abuse claims. These are likely to include hospitals, broadcasters, the church, judicial authorities and political bodies.

The detailed terms of reference are still being worked on and may be adjusted once the panel of the inquiry is identified. May is to reveal more when she addresses parliament later on Monday.

Cameron spoke to the home secretary over the weekend and agreed that no stone should be left unturned in finding the truth, and that any lessons from how organisations handled claims needed to be learned to understand society's failure as a whole to listen to the complaints of children.

The prime minister's spokesman pointed out that Kate Lampard, a former barrister and deputy chair of the Financial Ombudsman Service, had been given a role in overseeing scores of NHS inquiries into child abuse. The government said it was trying to balance the need for speed and being authoritative.

The shadow home secretary, Yvette Cooper, said the inquiry should be held in public, with experts chairing the inquiry.

The prime minister's spokesman said Cameron was not expressing a view on whether there had been an establishment coverup over allegations of child abuse by politicians in the 1980s, but said it was the responsibility of everyone to produce any relevant evidence.

The government also stressed that last year's review by the Home Office into how it handled thousands of documents about child abuse stretching back to 1979 had been conducted with no ministerial oversight. This is owing to long-established rules barring ministers from one party looking at the internal papers of a previous government led by a different party.

One issue remaining is the degree to which May was informed by the permanent secretary about the details of the 2013 review once it was completed.

The home secretary is under pressure to explain how her department lost or destroyed more than 100 files related to claims of organised paedophilia, as the government confirmed that details of four previously undisclosed allegations were handed to the police only last year.

The developments come after Lord Tebbit said he believed there may have been a political coverup of the 1980s allegations.

The Tory peer said: "At that time I think most people would have thought that the establishment, the system, was to be protected, and if a few things had gone wrong here and there that it was more important to protect the system."

Asked on Sunday on the BBC's Andrew Marr programme if he thought there had been a political coverup at the time, Tebbit said: "I think there may well have been. It was almost unconscious. It was the thing that people did."

The claims include allegations of abuse against the late Liberal MP Cyril Smith and allegations of paedophile activity at parties attended by politicians and other prominent figures. The missing Home Office material is reported to include details of officials, MPs and peers all implicated in child sexual abuse, including one Conservative MP at the time who was reportedly found with child abuse images but subsequently released by the police.

The Labour MP Tom Watson said the panel inquiry must be empowered to look at any papers inside government on how claims were handled. He said it was disturbing that there was a special branch file on Smith containing allegations of criminality that had not been followed up.

Watson said he understood why it might be necessary for the inquiry not to take evidence under oath since this might reduce the chances of prejudicing a future criminal case. But he said he believed the issue of missing files at the Home Office may prove to be a "red herring".