But in Brussels this afternoon the EU's Brexit negotiator said that there were no alternatives to the backstop, and that European Union leaders would not reopen the withdrawal agreement.
The report published today by the House of Commons" Exiting the EU Select Committee says that both the EU and UK have previously said that the matter of "citizens' rights' is a priority for them and that in the view of the MPs "The UK leaving the EU without a deal would cause real anxiety both for UK citizens living in the EU and EU citizens in the UK".
Senior Conservatives anxious that the government is drifting towards a no-deal Brexit, including the business minister Richard Harrington, had demanded that May set a two-week deadline for securing a settlement.
The Prime Minister said the government will support an amendment set to be voted on tomorrow that calls for the controversial Irish border "backstop" provision to replaced with "alternative measures".
Meanwhile, late on Monday evening it emerged a new Brexit plan had been put forward which is reportedly backed by members of both the Remain and Leave camps of the Conservative party.
Mr Brady said his amendment would give Mrs May "enormous firepower" if it is passed.
It will be down to Speaker John Bercow which proposals are selected for a vote later on Tuesday.
Her official spokesman said she aims to return to the Commons "as soon as possible" with a revised deal which will be subject to a "meaningful vote" by MPs.
Her announcement brought the prospect of the cancellation of Parliament's half-term recess a step closer, as the Commons is now due to rise for its 10-day break on 14 February.
"This is not a Brussels day, this is a London day".
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"What they said is, look, we voted remain, but we've had a vote: get on with it". Irish deputy PM Simon Coveney has said changes to the backstop proposal would not be acceptable.
"There's no negotiation between the United Kingdom and European Union - that's finished", she said. "A bit of realism is needed at this stage".
The paper also notes that in 2019 the EU is likely to be tied up with the approaching European elections in May, then a new Commission will be appointed in the autumn "which will absorb much of the focus on the new parliament".
"The practical consequences of the amendment would not be to rule out no deal but to delay Brexit", the spokesman said, adding that it offered "absolutely no positive suggestions" about how to resolve the impasse in parliament.
It has been reported that prominent Tories on both sides of the Brexit divide have hammered out a secret compromise created to bridge the gap between the warring factions within the Conservative Party.
In Tory ranks excitement is building over the Malthouse compromise, a plan co-ordinated by a minister Kit Malthouse proposed by leading soft dealer Nicky Morgan.
DUP leader Arlene Foster - whose 10 MPs prop up the minority Conservative administration in the Commons - said the plan provided a "feasible" alternative to the backstop.
The chances of the Commons derailing her plans were also heightened when Labour confirmed it will back a cross-party amendment to push Brexit day back from March 29 to the end of this year and put Parliament in the driving seat on the way forward.
The discussions involved herself, health minister Stephen Hammond, and solicitor general Robert Buckland from one wing of the party, and Jacob Rees-Mogg and Steve Baker from the Brexiteer side. At some point there has to be compromise on all sides in order to get a deal over the line.