The government is to publish plans later to give Stormont the power to vote on new EU rules if the border backstop comes into force after Brexit.
It comes as MPs get set to debate the withdrawal agreement ahead of a crucial vote in the House of Commons next week.
There will be five days of discussion on the terms of the UK’s withdrawal and future relations with the EU.
Cabinet Office Minister David Lidington said the plans being published were specific to NI.
The Democratic Unionist Party, who Mrs May relies on for her Commons majority, described the proposals a “meaningless” and of “no real significance”.
Sinn Féin said it is opposed to any possible “Stormont lock” mechanism.
“Anything that gives the DUP or extreme unionism any veto on progress in this society will be totally unacceptable,” the party’s John O’Dowd said.
“We are publishing today a set of Northern Ireland specific proposals that make clear the continuing place of Northern Ireland within the UK internal market and which will give the Northern Ireland Assembly – as we all hope it will be reconstituted – a veto over introducing any new areas of law and policy into that backstop,” Mr Lidington told the BBC.
The Northern Ireland Assembly has not met for two years, but Mr Lidington said there was a real desire to restore devolution.
“Talking to MPs and leaders of all political parties at Westminster, including the Northern Ireland parties, there is a wish to get the Northern Ireland institutions back up and working again,” he said.
“When I go to Belfast or Londonderry/Derry, what I get from people in Northern Ireland and from the business community and education leaders and others, is: ‘We want our devolved institutions back again because we want a voice for Northern Ireland and we want to see both communities in Northern Ireland represented in those power-sharing institutions’.”
Both the UK and EU have agreed there cannot be a “hard border” (for example physical checks such as cameras or customs posts) between Northern Ireland and the Republic of Ireland.
This led to the inclusion of the “backstop” in the withdrawal agreement, which means a last resort plan to keep the Irish border open no matter what happens – deal or no deal.
When the backstop was first agreed in December 2017 the joint statement from the UK and EU said that, if there was no agreement on how to prevent a hard border, then there would be no new regulatory barriers between Northern Ireland and the rest of the UK unless the assembly and executive agreed that Northern Ireland needed “distinct arrangements”.
Prime Minister Theresa May cancelled the original vote on the withdrawal agreement on 11 December.
She has since sought extra written assurances from the European leaders. The vote is scheduled to take place next Tuesday.
DUP MP Gavin Robinson said he did not think the proposals would lead to his party backing the withdrawal agreement.
“The withdrawal agreement says that even if there was disagreement in Stormont, even if Stormont decided that a proposal from the European Union wasn’t in the best interests of Northern Ireland, the UK government would legislate over our heads,” the Belfast East MP told BBC News NI.
“So in giving a consultative role, it doesn’t sound to me that the proposal that will be outlined today is going to be the sort of comfort that people were seeking.”
The DUP’s 10 MPs strongly object to the backstop, which they say curbs Northern Ireland’s freedom to strike trade deals and result in NI being treated differently from the rest of the UK.
Mr Robinson added: “If it [the backstop] was time limited, that would be a significant change and that would be progress.
“If there was the opportunity for unilateral withdrawal, that would be significant change and that would be progress.”
The Irish government has said it is willing to give further “written guarantees” to reassure MPs that the UK will not be “trapped” – although it has said the agreement cannot be changed.
Mr O’Dowd added that the backstop was “as good as it gets.”
“I’m of the view that there are elements of the DUP who now want a no deal, who would be perfectly happy with all that a no deal brings with it and have linked themselves to the extreme right wing of the Tory party,” he said.