If May is defeated in the House of Commons, it will be yet another blow to a prime minister whose authority has been challenged several times since she lost the Conservative Party's majority in an ill-judged election previous year.
The Lords amendment would have given parliament the power to decide what happened next, with the possibility of going back to the negotiating table or even staying in the bloc.
On the return of the EU Withdrawal Bill for consideration by MPs on Tuesday, the government won all the day's votes as it overturned a series of House of Lords amendments to the key Brexit legislation.
It was a message reinforced by her spokesman, who described the bill as "a vital piece of legislation for ensuring our statute book is ready for Brexit day and for delivering the smoothest possible exit from the European Union".
The SNP doesn't have any Lords because of an ideological disagreement with the concept of an unelected upper house - but it does have plenty of MPs in the Commons.
A bus passes as anti-Brexit demonstrators protest opposite the Houses of Parliament in London, Britain, June 11, 2018.
"The government can not demonstrate the flexibility necessary for a successful deal if its hands are tied midway through that process", Davis said. A victory for the "meaningful final vote" amendment would leave the government weaker in am upcoming round of talks with European Union negotiators in late June, and also weaken Theresa May's authority as leader.
The battle now moves to the House of Lords, where the government will formally reveal how much it has conceded in the wording of a new amendment expected on Monday or Tuesday.
Prior to the votes, the government suffered its first ministerial resignation over Brexit as Phillip Lee quit the Ministry of Justice so he could speak out freely.
A file photograph of justice minister Phillip Lees. Bercow said that no member of parliament should be threatened because over their views or vote on the Brexit bill.
Her concession to discuss the changes may mean lawmakers could have more power if she fails to secure a Brexit deal, possibly leading to a softer approach to Britain's divorce.
Theresa May has begged Tory rebels not to undermine her Brexit negotiating position on the eve of key votes in the Commons.
Nicky Morgan has been an outspoken critic of Brexit, but she backed the government following its concessions.
Also on Tuesday, the government successfully overturned an attempt to remove the date of Brexit from the face of the bill.
Rebels have said they will challenge May's plans to leave the customs union during votes on other bills, on trade and customs, which will be brought back to the house some time before July 24.
Britain is due to leave on March 29, 2019, and the bloc is frustrated with what it sees as a lack of firm proposals from the U.K about future relations.
Either way, the chances are increasing that Parliament will get a greater say over Brexit. A paper laying out the U.K. government position, due to be published this month, has been delayed because the Cabinet can not agree on a united stance.
May's government is divided between Brexit-backing ministers such as Foreign Secretary Boris Johnson who support a clean break with the European Union, and those such as Treasury chief Philip Hammond who want to keep closely aligned to the bloc, Britain's biggest trading partner.
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