A former Liberal Democrat minister is to demand new laws to make sure Britain's spying agencies can never intercept calls or email data without a specific warrant, arguing that Edward Snowden's revelations have exposed inadequate judicial oversight of surveillance powers.

David Heath, a former deputy leader of the House of Commons who left the government two months ago, is launching an attempt to tighten up the rules around interception of communications amid fears spies may be using legal loopholes to expand their powers.

He told the Guardian there were gaps in legislation governing the intelligence services that had to be closed, and called for the UK to follow America's example in looking at how to address the issue.

Heath will attempt to bring in his new bill from the backbenches, with cross-party support from MPs including Labour's Tom Watson, the Conservative former diplomat Rory Stewart, Caroline Lucas, of the Greens, Elfyn Llwyd, the Westminster leader of Plaid Cymru, and Simon Hughes, deputy leader of the Liberal Democrats.

There has been international outrage over revelations based on Snowden's leaks about the US tapping the phones of 35 world leaders and the power of the NSA and GCHQ to secretly access subsea cables carrying the communications of millions of internet users.

However, David Cameron and the heads of the intelligence agencies have accused the Guardian of endangering national security by publishing stories based on Snowden's leaks. Andrew Parker, the head of MI5, said the reports were a gift to terrorists, and the Tory MP Julian Smith has called for the Guardian to be prosecuted over the way it has stored and transported files.

The government insists the intelligence agencies always act within the law and only use surveillance powers with permission. The intelligence and security committee is examining oversight of the agencies. However, there is mounting pressure for a wider inquiry after Nick Clegg suggested there needed to be a closer look at the "unimaginable" power of spying technology.

On Wednesday the Commons will debate Heath's proposals after prime minister's questions, when the house is likely to be full. It is unlikely the legislation will progress without government support or parliamentary time.

He will ask for "leave be given to bring in a bill to amend the Regulation of Investigative Powers Act 2000 and the Intelligence Services Act 1994 to ensure judicial oversight of the use of material derived from British citizens by means of surveillance of telecommunications".

The former minister also wants to improve transparency in the operation of the investigatory powers tribunal and will suggest that the parliamentary intelligence and security committee is elected, rather than appointed by the prime minister.

Heath said the laws around surveillance were inadequate to deal with the potential for internet snooping by the security services. "By me saying this as a recent member of the government, I hope it makes the point that this is not a fringe issue," he said.

"We're very conscious of the need not to prevent the security services doing their work, but there are certainly gaps in the law. That much is clear. The fact America and others are trying to close those gaps suggest we are the odd ones out."

Heath said the spying agencies could already be exceeding their powers under the law and those that would have been given to them under the communications data bill, commonly known as the snoopers' charter, before it was blocked by the Liberal Democrats. "I don't know but the fact that we don't know is part of the problem," he said.