A former Labour cabinet minister has warned that GCHQ and Britain's other intelligence agencies appear to be undertaking mass surveillance without parliament's consent because the coalition failed to get the so-called "snoopers' charter" passed into law after Liberal Democrat opposition.

Nick Brown, a former chief whip who sat on the parliamentary committee scrutinising the draft communications data bill, said there was an "uncanny" similarity between the GCHQ surveillance programmes exposed by the US whistleblower Edward Snowden and proposals in the first part of the bill.

The communications data bill – dubbed the "snoopers' charter" by critics – would have given GCHQ, MI5 and MI6 much greater powers to gather and save information about people's internet activities but it was shelved in the spring amid Lib Dem fears that it intruded too much into privacy.

Brown, a Labour MP, said that it "looks very much like this is what is happening anyway, with or without parliament's consent" under GCHQ's secret Tempora programme, which was revealed by the Guardian in July in reports based on files leaked by Snowden. Tempora allows GCHQ to harvest, store and analyse millions of phone calls, emails and search engine queries by tapping the transatlantic cables that carry internet traffic.

Brown's remarks closely echo those of Lord Blencathra, the Conservative chairman of the committee and a former Home Office minister, who on Monday said GCHQ may be operating "outside the law or on the very edge of the law" because of Tempora.

But in parliament, Theresa May, the home secretary, told the home affairs select committee there was nothing in the Snowden files that changed the case for new laws giving the security services more powers to monitor the internet. She also described the Guardian's publication of the material as "damaging to the public interest" and repeatedly rejected the need for a debate on oversight of the intelligence agencies.

Afterwards, it emerged that Julian Smith, a Conservative MP, had written to the Metropolitan police asking the force to investigate whether the Guardian has broken the law by communicating intelligence information obtained from Snowden. The MP wrote to Sir Bernard Hogan-Howe to ask if offences had been committed under Section 58A of the Terrorism Act 2000 or the Official Secrets Act.

A Guardian spokesman said: "The high public interest in the stories we have responsibly published is evidenced by the debates, presidential review and proposed legislative reforms in the US Congress, throughout Europe and in Westminster. We're surprised that, once again, it is being proposed that terror legislation should be used against journalists."

Brown, an ally of the former prime minister Gordon Brown, also called for "grown-up cross-party discussions" to look at how to protect the privacy of individual citizens as the security services seek to expand their ability to monitor the internet.

"The similarity between part 1 of the proposed data communications bill and the events Mr Snowden is describing as already taking place is uncanny," Brown said. "It covers the same set of circumstances. The bill was trying to be permissive in that all material could be saved for a year. It now looks very much like this is what is happening anyway, with or without parliament's consent.

"I agree with Lord Blencathra's account of the evidence. I was sufficiently concerned not to accept part 1 of the [draft communications data] bill."

On Tuesday night, two other Liberal Democrat members of the joint committee also questioned why the Home Office did not reveal the extent of GCHQ's spying capabilities during the committee's inquiry, which concluded the bill carried a risk of "trampling on the privacy of citizens".

Lord Strasburger, a businessman, said nothing was mentioned about Tempora during two private "no holds barred sessions with the Home Office". "You have to wonder why, even in the secret sessions, none of the witnesses mentioned Project Tempora … It was highly relevant to our work and I believe that deliberate failure to reveal it amounts to misleading parliament," he said.

Last night, Jill Abramson, executive editor of the New York Times, mounted a defence of the ability of journalists at her own paper and at the Guardian to publish public interest stories based on the files leaked by Snowden.

"I think … that those articles are very much in the public interest and inform the public," she said on BBC's Newsnight.

On claims by MI5 director general Andrew Parker that newspaper reports were causing "enormous damage" to the fight against terrorism, Abramson said that there had been no proof of actual harm to security.

Julian Huppert, a Liberal Democrat MP on the committee, also said he

thought it was astonishing that May did not reveal the current

capabilities of GCHQ to the committee through their Tempora and Prism

programmes.

"Those of us on the committee were never told by any Home Office officials about the fact the data was already available," Huppert said. "In our report, we accused them of being misleading. It seems they were far more misleading than we could have realised at the time. Presumably, the home secretary knew that this was information they already had available, in which case she was not fully open with the committee. If she didn't know, that raises real questions about her role."

He said the government must reveal which ministers knew about Tempora after Chris Huhne, the former Lib Dem energy secretary, said he was not aware of the programme during his time in government.

"We know the cabinet was not briefed," Huppert said. "We have no idea who was. Was it just the prime minister? Was it a handful of others? Who made the decision not to tell other people? This is incredibly alarming. I hope we will be able to see proper debate and parliamentary scrutiny of this issue. We know that the security services play a very important role but they should operate with public consent."

• This article was amended on 16 October 2013. The original piece omitted a paragraph explaining that Julian Huppert was a member of the committee scrutinising the draft communications data bill. The paragraph has been reinstated.