Theresa May is to set out the details of an "independent and authoritative" inquiry into how the Home Office handled allegations of child abuse against politicians in the 1980s, amid mounting calls for a full public inquiry to run parallel with a criminal investigation.

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

The chancellor, George Osborne, said on Monday that the government would find an "independent and authoritative way" to get to the truth about any child abuse at Westminster.

Speaking from India, where he is touring with the foreign secretary, William Hague, Osborne said: "We need to get to the truth … we need to get to the bottom of what happened in many of our institutions, including potentially at Westminster."

It is understood May will address two issues: the Home Office's original response to the dossier alleging abuse in the 1980s, by appointing a QC to review it, and the wider issue of whether public bodies and other institutions have taken seriously their duty of care towards children.

Apart from the current police investigation, it means there will be three separate inquiries under way into the allegations.

A Home Office spokesman said an internal investigation by an independent expert last year found "13 items of information about alleged child abuse".

He said: "The police already knew about nine of those items, and the remaining four were passed to the police immediately. It is important that we do not pre-empt or prejudice any related police investigations."

The shadow home secretary, Yvette Cooper, has insisted there should be an independent review into how the Home Office handled the allegations at the same time as there being a criminal investigation into the claims.

"The priority has got to be the criminal investigations. At the same time you can have a full investigation into what happened within the Home Office," Cooper told BBC Radio 4's Today programme.

She said the internal review of how the Home Office handled the allegations had been inadequate.

Her comments come after Lord Tebbit said he believed there may have been a political cover-up of the child abuse allegations against politicians in the 1980s.

Tebbit 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 cover-up 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.

On Monday, Cooper said: "All we have had so far is that the Home Office did some kind of internal review, which they didn't publish, which was a summary which was put on to the internet last year which nobody was notified about.

"We discover almost a year later that over 100 files were missing. That wasn't properly looked into and that's why you need a serious investigation that looks not just at the Home Office but much more widely into what happened and why these allegations were not pursued the first time round. You can do that at the same time as a criminal inquiry."

Cooper insisted that such an inquiry would not be in contempt of court.

"There are other cases where there have been parallel investigations into what went wrong in institutions alongside criminal investigations. Of course there may be times when information needs to be passed across to the police.

"Most important in the long run is pulling together all of these different investigations and looking at what does this mean for our child protection system in the future, because that is where I have the greatest concern. This is not just about history, this is about the need for proper strong systems of child protection for the future, so that we get both justice for victims in the past but also a system that is strong enough to protect young people going forward."

Sir Peter Bottomley, who served as a Tory minister during the mid-1980s, said he was sceptical of a cover-up but thought there was a need for a public inquiry.

He told Today: "The people in charge of social services in the Department of Health, [including a former] director of social services Norman Warner, Sir William Utting, Herbert Laming, Denise Platt, were all public servants and would not have been involved in any kind of cover-up whatsoever. My knowledge of the Crown Prosecution Service and the DPP [director of public prosecutions] is that they wouldn't have been involved in any organised thing either. We do need the inquiry, [but] we do need for it not to interfere with the possible gathering of evidence and fair trials."

Bottomley said paedophile networks were not taken seriously enough at the time.

"Sometimes MPs are too busy, sometimes departments are overloaded, but it was also quite clear, looking at both Conservative and Labour governments, that things like the paedophile information exchange weren't taken as seriously as they ought to have been, partly because people couldn't believe it; partly because there may have been one or two senior people in, say, the police, or say, some other forms of public life. But the idea that there could have been some organised group who could have stopped things happening is, I think, imagination."