Parliament's intelligence watchdog should investigate David Cameron's role in asking the Guardian to destroy or surrender leaked secret NSA documents, Yvette Cooper has said.

The shadow home secretary made her call after the Daily Mail and the Independent reported that Sir Jeremy Heywood, the cabinet secretary, made the request to the Guardian on the instructions of the prime minister.

Cooper told BBC Radio 4's Today programme: "We don't know what was on the [disk drives] or what the material was that the government was pursuing. Clearly the government does have a responsibility to protect national security. However, I think this may be another area where an inquiry by the intelligence and security committee [ISC] may be the right way forward in terms of this particular case and what the prime minister's role was."

Alan Rusbridger, the editor of the Guardian, disclosed on Monday night that a "very senior government official claiming to represent the views of the prime minister" asked him to return or destroy all the NSA documents leaked to the paper. The Guardian agreed to destroy two hard drives last month in the presence of two security experts from Britain's GCHQ eavesdropping centre after the government threatened to take legal action.

Rusbridger told officials that the Guardian would continue to report from the leaked documents because it had backup copies in the US and in Brazil. Glenn Greenwald, the Guardian journalist who received the documents from the US whistleblower Edward Snowden, lives in Rio de Janeiro.

The Guardian declined to name the official who contacted Rusbridger. But the Daily Mail, the Independent and the BBC said that Heywood had contacted the Guardian editor, acting on the authority of the prime minister.

A government source denied to the Independent that Heywood had acted in a "threatening" manner. The source said: "We had a mature conversation." A Guardian spokeswoman declined to comment on the Independent's disclosure, telling the BBC: "We're not going to comment on this."

In her BBC interview, Cooper suggested that the government may have acted in an evasive manner after the nine-hour detention of David Miranda, Greenwald's partner, at Heathrow airport on Sunday. Miranda was detained under anti-terror laws as he flew home to Rio from Berlin via London.

During his trip to Berlin, Miranda met Laura Poitras, the US film-maker who has been working with Greenwald and the Guardian. Officials confiscated Miranda's mobile phone, laptop, camera, memory sticks, DVDs and games consoles.

Cooper said: "I have two concerns about this case. The first is whether or not it was appropriate or legally justified to use terrorism powers in this case when there were other legal avenues that could have been pursued. The second was whether the home secretary and the government have been evasive about their role in this process, which has rather had to be dragged out of them. We still don't know the full position."

The shadow home secretary questioned the use of schedule 7 of the Terrorism Act 2000 to detain Miranda after Lord Falconer of Thoroton, the former lord chancellor, said there was no legal basis under the act to hold him. Falconer told the Guardian that police had the right to detain anyone, even when they do not suspect them of terrorism. But they have to assess whether the person has been involved in the commission, preparation or instigation of terrorism. "Plainly Mr Miranda is not such a person," Falconer said.

Sir Malcolm Rifkind, the former foreign secretary who chairs the ISC, said the use of anti-terror laws to detain Miranda was a "sensitive issue" that should be investigated. But he told the Today programme: "This was not about embarrassment to the government. The documents which Snowden stole from the National Security Agency are documents some of which deal with how the intelligence agencies get access to terrorist information through interception of mail or phone messages. That is something potentially relevant to terrorists and therefore it is not a question of embarrassment to the government."

Rifkind was strongly supportive of the way in which the government sought the return or destruction of the leaked NSA documents. "I think Mr Rusbridger, in the article he wrote about the destruction of his hard disks, is on relatively weak ground. He clearly did not dispute that he had no legal right to possess the files or the documents. The question was whether he handed them back to the government or whether they were destroyed. He chose the latter option.

"Clearly if he thought that what he was doing was perfectly lawful, that he was perfectly entitled to have these documents, he would have told the cabinet secretary – or whoever it was – to go and get lost and take me to court. But he didn't do that. He knew perfectly well that if you have in your possession documents which were originally stolen you are on pretty dodgy ground."