David Cameron has encouraged a Commons select committee to investigate whether the Guardian has broken the law or damaged national security by publishing secrets leaked by the National Security Agency whistleblower Edward Snowden.

He made his proposal in response to a question from former defence secretary Liam Fox, saying the Guardian had been guilty of double standards for exposing the scandal of phone hacking by newspapers and yet had gone on to publish secrets from the NSA taken by Snowden.

Speaking at prime minister's questions on Wednesday, Cameron said: "The plain fact is that what has happened has damaged national security and in many ways the Guardian themselves admitted that when they agreed, when asked politely by my national security adviser and cabinet secretary to destroy the files they had, they went ahead and destroyed those files.

"So they know that what they're dealing with is dangerous for national security. I think it's up to select committees in this house if they want to examine this issue and make further recommendations."

The prime minister's spokesman refused to elaborate on what Cameron meant by the issue of the Guardian disclosures being examined by a select committee.

There are as many as four committees that might take up David Cameron's suggestion, including the culture select committee, the home affairs select committee, the defence select committee and the intelligence and security select committee.

The ISC largely meets in private but is due soon to meet the leaders of the spy agencies in public and it is certain that the issues raised by the Guardian, including the impact on national security, will be discussed.

Cameron did not follow calls by the backbench Tory MP Julian Smith for the police to prosecute.

Fox had asked the prime minister whether "we can have a full and transparent assessment about whether the Guardian involvement in the Snowden affair had damaged Britain's national security".

He said it was bizarre of the Guardian to claim that leaving Britain's secrets and security personnel more vulnerable was opening a debate about the accountability of the security services.