Theresa May is facing another major rebellion over Brexit amid a cross-party move to kill off any attempt to crash out of the EU without a deal.

A group of MPs and peers has been carefully crafting new laws that will hand parliament guaranteed powers to soften any deal and send the government back to the negotiating table.

Tory sources already think there is enough support in the Commons for the plan, which they believe would end the threat of a “no deal” outcome. The move is being described by senior figures as one of the most significant amendments of the Brexit process.

Ministers have previously warned that should the deal negotiated with the EU be voted down by parliament, Britain could simply leave the bloc with no agreement. However, the cross-party group’s proposal ensures that if the agreement were voted down, parliament could alter it and ask the government to reopen EU talks.

The measure is expected to win a three-figure majority in the Lords on Monday and Tory Remainers are confident it will not be overturned in the Commons next month. The government is already being urged simply to accept the measure.

It could open the way for parliament to back staying in the EU’s customs union and single market –something currently ruled out by the prime minister.

It comes with rebel Tory MPs beginning to believe that there could now be a Commons majority for staying in the customs union and its single market, through membership of the European Free Trade Association.

One well-placed Tory said there were “a few in cabinet who now see it as a possible solution”. However, the cabinet’s Brexiters are preparing a big push this week against any suggestion Britain should remain inside the customs union, which would make it impossible for the UK to negotiate its own trade deals.

While Brexit minister David Davis indicated last week that parliament would have a real “meaningful vote” over the Brexit deal on offer, both peers and Remain MPs want to enshrine the guarantees in law. Labour regards the proposal as extremely significant because it would allow its MPs and Tory rebels to vote against the agreement on offer without the fear of triggering a disastrous “no deal” outcome. Keir Starmer, Labour’s shadow Brexit secretary, told the Observer: “This is one of the most important amendments of the entire Brexit process – and indeed of the parliament. We have always been clear that the vote must be truly meaningful. It cannot simply be a take-it-or-leave-it choice as the prime minister has suggested.

“This amendment, which has cross-party support, would provide a safety net in the Brexit process. It would remove the possibility of a No vote leading to a no-deal. It would bring back control to parliament.”

Facebook Twitter Pinterest Brexit minister David Davis has insisted parliament will have a ‘meaningful vote’ on the final deal. Photograph: Steve Parsons/AFP/Getty Images

The measure is being presented in the Lords by a group of peers led by former Tory minister Lord Hailsham.

However, the government describes the measure as a meaningless attempt to frustrate Brexit.

Lord Callanan, the Brexit minister in the Lords, said: “Labour’s flawed amendment seeks to tie our hands by inserting false deadlines and shifting the power to negotiate from government to parliament. It asks for meaningless votes on the deal before the deal is done. Those who want to overturn the referendum call this the ‘no Brexit’ amendment.

“The Conservatives are taking the scrutinising role of parliament seriously, to improve an essential piece of legislation. Labour are using it to frustrate Brexit.”

Senior Labour MPs, meanwhile, are directly appealing to their counterparts in the Lords to back attempts to keep Britain in the customs union and single market. Chuka Umunna, Heidi Alexander, Alison McGovern and Chris Leslie have written to Labour peers urging for their help.

“We are writing to ask for your support for an amendment which requires the government to make the UK’s continued participation in the European Economic Area (EEA) an objective in the Brexit negotiations,” they write. “The EEA provides a platform on which we can continue to be part of the EU single market if Brexit occurs, with the protection for workers, consumers and our environment that our party believes in.”

Another Labour MP, meanwhile, is attempting to engineer the first Commons vote on whether or not the final Brexit deal should be put to a second referendum. Gareth Thomas, the former trade minister, will introduce a backbench bill next month, which will trigger a non-binding vote on the issue.

“A people’s vote would give the British people the chance to take back control over Brexit from a small cabal of Leave ideologues and instead, on the biggest question facing our country, determine together our country’s economic, social and cultural future,” he said.