However, sources said there was immediate disagreement over whether the proposal would give the United Kingdom the power to unilaterally call time on the backstop, with both the home secretary, Sajid Javid, and the foreign secretary, Jeremy Hunt, arguing that it must.
The Sun newspaper reported Tuesday that Raab and Foreign Secretary Jeremy Hunt would use the cabinet meeting to warn that any deal with the European Union that traps Britain within the bloc's orbit would never be approved by MPs.
The focus was on the way in which the United Kingdom could leave the EU customs union.
In a possible signal the meeting went well, Brexit secretary Dominic Raab gave a "thumbs up" to reporters as he left No10, she added.
Therefore in order to get the current iteration of the backstop through cabinet, the PM must persuade ministers that the EU has no intention to trap the United Kingdom in a customs union in perpetuity.
"The Taoiseach (Prime Minister) indicated an openness to consider proposals for a review, provided that it was clear that the outcome of any such review could not involve a unilateral decision to end the backstop", the statement said.
"In order to ensure that the backstop, if ever needed, would be temporary, the Prime Minister said that there would need to be a mechanism through which the backstop could be brought to an end".
Some at the highest levels of government fear that, unless progress is agreed by Tuesday when the May sees her senior ministers and parliament breaks for recess, the cabinet may not have a direct input before a summit announcement is made.
The Downing Street spokesman said the cabinet "needs more time" to consider mechanisms to ensure Britain can not be bound to the EU.
Mr Gauke told a Channel 4 Brexit debate show: 'If we leave on no-deal terms there's no good shying away, it will be very bad for us economically.
"The Prime Minister added that the United Kingdom wanted to see quick progress and that both the United Kingdom and the European Union were working hard to achieve an agreement", they added.
"We are coming to a stage where the markets believe that a deal may be on the cards and sentiment is not that negative towards sterling as it was a few weeks ago", said Manuel Oliveri, an FX strategist at Credit Agricole in London. In the 2016 referendum, they voted 52-48% in favour of Brexit.
Ireland's Europe minister Helen McEntee told RTE on Tuesday that the proposal could pave the way to a deal.
His comments came after Labour MP Christian Matheson raised concern over the impact of Brexit on public services.
Mr Matheson, speaking in Treasury questions, said: 'The Government's own figures demonstrate between a 2% and 8% hit on the broader economy on Brexit, so isn't it the case that there is no form of Brexit that won't have a massive impact on the public finances and therefore on public services?'