But Mrs May is facing a fresh showdown with pro-European MPs in her own party after the House of Lords overwhelmingly backed plans to give parliament a "meaningful vote" on the Government's final Brexit deal.
Sometime after 1400 GMT, the House of Lords will debate different proposals for a "meaningful" vote - the role parliament will play if lawmakers reject a deal that May negotiates with the European Union, or if she fails to agree any exit deal.
Peers voted 354 to 235 for an amendment to the bill giving MPs a "meaningful vote" on the final Brexit deal.
Grieve and a cohort of potential Conservative rebels with enough votes to defeat the government want parliament to have more influence over the next steps than ministers are prepared to give them.
Davis and Prime Minister Theresa May have argued that they can not accept anything which gives parliament the power to bind their hands in negotiations with the European Union, or opens the door to lawmakers overturning the result of the 2016 Brexit referendum.
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But Lords' leader Baroness Evans of Bowes Park said the amendment had "flaws" and insisted the Government had come back with a "fair, practical and constitutionally sound offer" to meet the demands of peers and MPs. "But there is a risk it will happen, and if we have no deal at the very end it is a serious crisis", Grieve said.
British Prime Minister Theresa May and her advisers were Tuesday working on a strategy as they prepare for a vote in the House of Commons in just 24 hours following a crushing defeat in the unelected House of Lords.
"We could collapse the government, and I assure you I wake up at 2:00 am in a cold sweat thinking about the problems that we have put on our shoulders", he said.
A spending pledge for Britain's public health service by May has also irritated some pro-EU lawmakers, who question the government's assertion that the new funding is part of a "Brexit dividend" - money saved by leaving the bloc.
"I would say to all of my colleagues: vote in a way to make sure the Government can get the best deal for the United Kingdom, because that's what the people of the United Kingdom need and deserve".
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Lord Hailsham said: "The government's amendment not only fails to deliver the promised meaningful vote".
However, the eventual government amendment did not go as far as one drafted by Dominic Grieve, the Tory former attorney general.
Leading backbencher Brexiteer Jacob Rees-Mogg said that the meaningful vote was being used to "obstruct Brexit" and "thwart the referendum result".
The government then has two weeks to return to the Commons with a statement on how the government plans to proceed.
"MPs now face a decisive vote on Wednesday to guarantee Parliament has a proper role in the Brexit negotiations and to take the threat of no deal off the table once and for all".
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