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 a year ago.
The parts of his amendment which he expects to be taken forward by ministers provide a mechanism by which Parliament has to be consulted by the end of November in the event of no deal or if a proposed agreement is rejected, he said.
Facing the prospect of losing a vote on a crucial amendment to the government's flagship Brexit legislation - which was created to empower parliament to vote down the final deal without risking a "no-deal" exit from the bloc - ministers intervened with a concession at the 11th hour even as MPs were wrapping up debate on the controversial measure.
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.
In London, Parliament is considering crucial pieces of law that need to be passed in order for the U.K.to leave the bloc without causing legal chaos. Sources have said the government is happy to implement the first two points and will discuss the last one.
"Thirdly, we must under all circumstances respect the result of the referendum".
"The government can not demonstrate the flexibility necessary for a successful deal if its hands are tied midway through that process", Davis said.
The Government's compromise is that a minister would come to the House within 28 days of a deal being rejected to tell MPs what will happen next, but an amendment tabled by former attorney general Dominic Grieve would insist on a binding Commons motion.
Remain-supporting Conservative MPs had threatened to back an amendment to the bill which would have given parliament a more widespread veto.
While in the end, only two Tory MPs - Ken Clarke and Anna Soubry - voted against the government, there were clashes over how much of a say Parliament should get as the United Kingdom leaves the European Union, with one side accusing the other of trying to "wreck" Brexit - and being accused in turn of being "zealots" who wanted to sideline Parliament.
Phillip Lee, who resigned this morning, gave an impassioned speech from the "naughty corner" on the backbenches - flanked by Remainers including Bob Neill, Nicky Morgan, receiving congratulations for his decision by Soubry and Sarah Wollaston.
The change reduces the likelihood that Britain could leave the European Union without a deal if it does not like the divorce terms.
Theresa May has begged Tory rebels not to undermine her Brexit negotiating position on the eve of key votes in the Commons.
The bill underpins the government's Brexit strategy.
However the Conservative politician stated the Government would still need to compromise or face another showdown in Parliament when the bill returns at a later stage of the legislative process.
"Whatever we do, we're not going to reverse that (decision to leave the EU)", David told BBC radio.
Another flashpoint could come when lawmakers vote Wednesday on an amendment seeking to keep Britain in a customs union with the EU.
Two years after Britain voted to leave the European Union, and eight months before it's due to leave on March 29, 2019, the bloc is frustrated with what it sees as a lack of firm proposals from the United Kingdom about future relations.
Brexit protesters outside Parliament House.