With the Conservatives on course for a drubbing in Thursdays European elections, the prime minister hopes the results will focus the minds of her own MPs and persuade them to support the long-awaited withdrawal agreement bill (WAB).
Despite the collapse of cross-party talks with Labour, ministers hope some of the measures discussed can still be bolted on to the bill, as part of what May has called a new, bold offer to MPs across the House of Commons.
The Brexit secretary, Steve Barclay, told Skys Sophy Ridge on Sunday: Weve been in talks with the Labour party, weve been exploring issues around workers rights, environmental standards, what commitments can be given to parliament in terms of the next phase of negotiations.
The development secretary, Rory Stewart, told the BBC the two sides in the talks had been half an inch apart, and added the issue of trading with Europe to the list of areas where MPs should expect to see movement.
Proposals are expected to include separate legislation to ensure parliament is given a vote on whether to adopt any improvements to workers rights introduced by the EU27 in future though that would fall short of Jeremy Corbyns call for changes to be automatically adopted.
The government is also keen to offer fresh reassurances to the Democratic Unionist party, which has been resolutely against Mays deal and is particularly concerned about the risk of regulatory divergence between Northern Ireland and the rest of the UK.
I still dont get any sense that its going to pass but we might as well throw the kitchen sink at it, said one government insider. But another said: There is no offer as far as we can see: its just the WAB.
Several cabinet ministers, including Barclay, are likely to press for the government to ramp up no-deal Brexit preparations, in case Mays deal is defeated yet again.
Members of parliament do need to face facts, and if the deal were not to go through then there are only two alternatives you either leave with no deal or you revoke, he said.
If parliament wont back a deal I do think we need to bring forward our preparations to mitigate no deal, because we will need to use the additional time we have, and we need to move at pace to do so.
Cabinet ministers keen on a softer Brexit prefer the idea of holding a series of votes in parliament before the bill is tabled, a process which could reveal a majority for a customs union though that was not the outcome the last time indicative votes were held.
In an article in the Sunday Times, May said she would be discussing with ministers whether holding votes in parliament to test support for possible solutions would be a useful prelude to MPs considering the legislation. A Downing Street source said the idea remained under consideration.
A document tabled in the talks with Labour and leaked last week suggested the government had proposed holding indicative votes in the next few days including the offer of a free vote on whether to submit any deal to a referendum.
As well as the substance of the governments bold offer, May must decide on when the crunch vote on the bill will be held.
Downing Street has committed to the week beginning 3 June; but with Donald Trump and a string of other leaders visiting the UK that week to mark the anniversary of D-day, timing is tight.
If the government hopes to hold the vote at the start of the week, before the US president arrives, it would face pressure to publish the bill this week, before MPs disappear for a Whitsun recess.
But that could amplify objections to the governments policy as voters prepare to head to the polling stations for European parliament elections.
Several of the Brexiters who were won over to supporting the government at the last meaningful vote have signalled they could oppose the bill, against the backdrop of a vociferous campaign against Mays deal from Nigel Farages Brexit party.
Leadership frontrunners Boris Johnson and Dominic Raab have both declined to say whether they would support it again, making it even harder for May to win a majority. She was defeated last time by a majority of 58.
EU27 ministers have set the government a deadline of 31 October to pass the legislation necessary to enact Brexit, or come up with an alternative policy.
If Mays deal falls in early June, much of the summer is expected to be taken up with choosing the next Tory leader who will then have to set out their own alternative policy.