A government bid to find a compromise to win over rebels before the Commons Brexit vote has run into criticism.
A Tory backbench amendment – understood to have No 10’s backing – offers MPs more of a say over the contentious issue of the Northern Ireland backstop.
DUP leader Arlene Foster dismissed it as “legislative tinkering” while Tory Brexiteers said it was “desperate”.
Many MPs have expressed concerns about the backstop, aimed at preventing a “hard border” on the island of Ireland.
It would mean Northern Ireland staying aligned to some EU rules, which many MPs say is unacceptable.
The UK would also not be able to leave the backstop without EU agreement.
The withdrawal deal negotiated between the UK and EU has been endorsed by EU leaders but must also be backed by Parliament if it is to come into force.
MPs will decide whether to accept it next Tuesday, but dozens of Tories are expected to reject it, as will the DUP, whose support keeps Mrs May’s government in power.
Will May’s suggestion win over any Brexiteers?
By Laura Kuenssberg, BBC political editor
What Theresa May sketched out on Thursday was the idea of allowing MPs to choose when and if they want to go into the controversial “backstop” – the insurance policy against a hard border on the island of Ireland.
Clearly, to try to get some angry Brexiteers to change their minds, the PM is trying to give a sense that they might have more of a say.
They could, as the agreement already suggests, just extend the “transition period”, giving the two sides longer to come up with a free trade deal that would mean the dreaded backstop is never used.
It’s not surprising that MPs would have a vote on that. But Number 10 clearly hopes it will give some grumpy MPs a sense that they will have more of a say, introducing another layer of decision-making so that the backstop can be avoided.
The prime minister has suggested that MPs could be “given a role” in deciding whether to activate the backstop, and on Thursday night, a Tory backbench amendment was laid down intended to do that.
The amendment – which is understood to have government support – would also give the devolved administrations – particularly the Northern Ireland Assembly – more say in the process, and press the UK and EU to agree a future trade deal within a year of the implementation period ending.
Former Northern Ireland minister Hugo Swire tabled the amendment along with Bob Neill and Richard Graham.
Mr Graham said: “What we are trying to achieve is something that gets a lot of support from colleagues and that the government, we hope, will take forward because it will make a real difference to the vote.”
But Conservative Brexiteer Steve Baker said: “Giving Parliament the choice between the devil and the deep blue sea is desperate and will persuade very few.”
And one senior source from the Conservative European Research Group told the BBC it was “transparent and risible”.
And DUP Leader Arlene Foster tweeted: “Domestic legislative tinkering won’t cut it. The legally binding international withdrawal treaty would remain fundamentally flawed, as evidenced by the attorney general’s legal advice.”
Cabinet ministers will travel around the UK on Friday as Theresa May continues to seek support for her Brexit deal.
Five days before MPs vote on the deal negotiated with the EU, Philip Hammond and Matt Hancock will be among those trying to sell it to the public.
The chancellor will visit a school in Chertsey, Surrey, while the health secretary will go to a hospital in Portsmouth on Friday.
Cabinet Office Minister David Lidington will meet small business leaders in Belfast and Scottish Secretary David Mundell is due to speak to employers in Glasgow.
Around 30 ministers in total will join the push to gain support for the withdrawal agreement.
Theresa May said: “I’ve been speaking to factory workers in Scotland, farmers in Wales and people right across the country, answering their questions about the deal and our future.
“Overwhelmingly, the message I’ve heard is that people want us to get on with it.
“And that’s why it’s important that ministers are out speaking with communities across the UK today about how the deal works for them.”
But in an article in the Guardian, Labour Leader Jeremy Corbyn said the deal represented a “monumental and damaging failure for our country” and represented “a worst of all worlds deal, that works for nobody, whether they voted leave or remain”.
He confirmed that, if the deal was rejected and Labour’s preferred outcome – a general election – was not on offer, “all options” remained on the table, including “the option of campaigning for a public vote to break the deadlock”.