Theresa May’s Brexit preparations could be thrown into fresh difficulty on Tuesday, with Labour poised to support a backbench amendment tabled by Yvette Cooper that could restrict the government’s tax powers unless a no-deal Brexit is taken off the table.

The government may be forced to concede after the Labour frontbench suggested it was likely to whip its MPs to back the cross-party amendment to the finance bill, significantly increasing its chances of success in the Commons. Around a dozen Tory MPs have signalled their intention to back the amendment.

Downing Street and Treasury sources suggested on Monday night that the material effect of the amendment on no-deal preparations for tax administration would be inconvenience rather than disaster.

The instinct in No 10 is to ride out the possible damage and push ahead with the vote on the finance bill on Tuesday, though it is undecided if Tory MPs will be whipped to oppose the amendment or if the government will concede to avoid a defeat.

Yet the amendment’s supporters are also understood to see it as a vehicle to demonstrate the strength of parliamentary opposition to no deal. MPs are also beginning preparations to target other bills with similar amendments that would give parliament more levers to avert a no-deal Brexit.

Downing Street announced on Monday that there were seven bills that have to be implemented in order to provide a smooth exit, including trade, agriculture, healthcare, financial services, fisheries and immigration, as well as legislation for the withdrawal agreement.

Writing for the Guardian, Cooper said there was a growing danger that “brinkmanship, political paralysis, siren voices” could lead the UK over a Brexit cliff-edge.

“Just because most people don’t want something to happen, doesn’t mean it won’t,” she said. “So it is with a no-deal Brexit. I believe the majority of MPs, ministers and most people across the UK are opposed to the self-inflicted damage to our economy and security that would result from crashing out of the EU.”

The amendment has the backing of a number of Labour and Conservative select committee chairs, including Tories Nicky Morgan and Sarah Wollaston, Labour’s Hilary Benn and Frank Field, the pro-Brexit chair of the work and pensions select committee. Former minister Nick Boles and Tory grandees Sir Oliver Letwin and Nicholas Soames have also given their support.

The House of Commons Speaker, John Bercow, must rule on whether to accept the amendment ahead of the finance bill debate on Tuesday but it is understood to have a high chance of selection if the debate goes ahead.

Cooper said the amendment had support from MPs on both sides of the Brexit debate, including from those committed to backing May’s deal and those opposing it. “It doesn’t avoid the difficult debates and disagreements ahead over the best way forward, but it at least gives us the chance to rule out the worst outcome,” she said.

The amendment to the finance bill would be attached to a clause designed to give the government the power to keep tax law working in the event of a no-deal Brexit.

If it were passed, it would mean that clause would only be allowed to come into force if there were either a Brexit deal, a decision to extend article 50, or a vote in the Commons specifically approving a no-deal Brexit. It is this format of three options that MPs may later decide to attempt to apply to other bills going through the Commons.

Morgan said it was time for parliament to create a mechanism to stop no deal by default. “Many of us have been clear that parliament will not allow a no-deal situation to unfold, and with less than 12 weeks to go until 29 March it is time for parliament to show our opposition to a no-deal exit,” she said.

Experts and Whitehall sources are divided about the effectiveness of the amendment. One Treasury source said: “It makes things more difficult but compared to the general effect of no deal, it adds to problems but it is quite a long way down. It does not stop no deal.”

Nikki da Costa, the former Downing Street head of legislative affairs who has extensive experience navigating the difficult parliamentary arithmetic since 2017, said the Treasury would have to weigh up if the powers could be reinstated via another route.

“The easiest route is to see if those powers really are essential and how quickly they may be needed,” Da Costa said, saying she believed the actual effect to be “relatively minor”.

“If those amendments, for example to EU references in tax law, aren’t really problematic, and you can afford not to amend for some time after exit day, then you might just accept the amendment and find another legislative vehicle at a later date.”

Other efforts are also under way to demonstrate the extent of parliamentary opposition to a no-deal Brexit. Some of the 200 MPs who signed a letter organised by Tory ex-minister Dame Caroline Spelman and Labour’s Jack Dromey, calling for May to rule out a no-deal Brexit, are set to meet the prime minister on Tuesday.

Separately, Labour is expected to face renewed attempts by grassroots party members to back a second referendum this week. Left-wing activists from the anti-Brexit group Another Europe is Possible have organised for members for more than two hundred 200 local parties to submit pro-referendum motions for the party’s national policy forum on Wednesday.

The party’s policy cannot be officially changed again until its 2019 conference, which will happen after the UK is expected to have left the EU, but organisers hope that the sheer number of motions submitted for debate at the policy forum will have heavy symbolism in the crucial weeks ahead.