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.
Solicitor General Robert Buckland said the government remained "open-minded", but this may or may not result in it coming forward with new proposals in the coming days.
If some of its leave-supporting lawmakers choose to vote against the amendment, the government could avoid defeat altogether.
The pound traded higher against the euro and the dollar after the votes.
Shadow Brexit Secretary Keir Starmer commented: "This vote was about ensuring parliament was given a proper role in the Brexit negotiations and that we avoid a no deal situation, which is becoming more likely with the divisions at the heart of this government".
Her government is most vulnerable over an amendment, introduced by the upper house of parliament, to change the so-called "meaningful vote" on any final Brexit deal by handing the lower house more power to set the "direction" of the government if it rejects the agreement.
The House of Commons voted 324 to 298 to defeat an amendment to the EU (Withdrawal) Bill which would have removed her government's power to unilaterally walk away from talks with Brussels.
Outside parliament, around 60 protesters called on parliament to stop Brexit, saying they were cheated by the 2016 campaign to leave the European Union which they suspected had been influenced by Moscow.
The Government made a series of late concessions to backbench MPs who were minded to rebel behind closed doors, meaning the true nature of what has been agreed is not yet clear.
Remainer Stephen Hammond said a group of potential rebels - believed to number 15-20 - received assurances from the PM moments before the key vote.
British Prime Minister Theresa May faces a showdown in parliament with lawmakers later on Tuesday who want a "meaningful vote" on an eventual Brexit deal and to set the government's "direction" if the house rejects the agreement.
‘No relationship with their father is harmful’: Angelina Jolie may lose custody
A judge in the former couple's case believes the children " not having a relationship with their father is harmful to them ". He is responsible for returning them back to Jolie in London on July 29, in addition to paying for their transportation.
Davis on Tuesday told the BBC, "that was the decision of the British people. and whatever we do, we're not going to reverse that".
The concession means MPs could now be handed the power to seek to prevent Britain crashing out of the European Union without a Brexit deal.
"This needs to be resolved", Andrew Bridgen, a pro-Brexit Conservative lawmaker, told Reuters.
The debate, which lasted for almost three hours, was split down the usual non-partisan lines that have emerged as a result of Brexit, with the likes of Labour's Kate Hoey and John Mann saying they would back the Conservative government, while Tories including Ken Clarke and Anna Soubry spoke in favour of Grieve.
The problem for the prime minister is that she can't keep both sides happy.
One of the key points of difference between the Prime Minister and the rebels is a Lords amendment which states the Government must seek to negotiate a customs union with the EU.
There is little May can do.
Ms. May's minority government relies on the support of Northern Ireland's Democratic Unionist Party for a slender working majority in the 650-member Commons.
But as time ticks by, she can no longer kick decisions down the road, increasingly under pressure from European Union negotiators to come up with detailed positions not only on customs, but also on the wider trade agreement and governance.
Earlier May suffered a setback when junior justice minister Phillip Lee, who has always been critical of Brexit strategy, resigned and said he would vote against the government.