Mr Davis said his officials are still working on a United Kingdom version of the backstop.
The amendment, which was supported by 19 Tory rebels, means parliament could now force negotiators back to Brussels if they do not approve the terms put to them in October.
Meanwhile, there is mounting pressure on the United Kingdom from another quarter as the EU's chief Brexit negotiator said that Britain and the European Union must reach an agreement on the key points of the Northern Ireland border issue by June.
Both sides hope to negotiate the UK's withdrawal agreement by this October in order to give the United Kingdom and European Parliaments enough time to debate and vote on it before the moment of departure.
The government on Monday night suffered yet another defeat in the House of Lords, as peers voted against the EU Withdrawal Bill for the seventh time.
The amendment, known as Clause 49, proposes that Parliament can determine the Government's course of action if the Commons rejects.
Asked what the alternative to a second referendum would be if Parliament voted against the deal reached with Brussels at the end of the negotiations, he replied: "If in the end we have an agreement that the Government wants but Parliament won't agree to, we have the possibility of leaving the European Union without an agreement".
Brexit Secretary David Davis last week admitted that the government would have the option of asking for an extension to Article 50 negotiations, should Parliament request it.
But the Lords heavily voted for Parliament to be given a more decisive say on the outcome of the Brexit negotiations.
"We will now consider the implications of the House of Lords' decision".
"That Enabling Bill transferred the democratic right from the parliament into the hands of one man, the chancellor", he said.
The Lib Dems are pushing for an amendment which would make the PM hold a referendum on the final Brexit deal.
Theresa May is set for a clash with backbench Brexiteers over plans to water down post-Brexit immigration controls.
"This amendment would weaken the UK's hand by giving Parliament powers to instruct the government to do anything as we seek a deal".
Euro zone manufacturing cools off in April but remains warm
According to the PMI index updated on Wednesday, manufacturing performance increased to 51.6 in April 2018 from 51 in March. Adding to signs that global demand is softening, the new export orders subindex declined for the first time in 17 months.
NSYNC receives star on Hollywood Walk of Fame
The star is the 2,636th since the completion of the Walk of Fame in 1961 with the first 1,558 stars. The crowd applauded Bass. "This is so surreal", Timberlake said at the ceremony .
There is a BJP storm, not a wave, in Karnataka
Vishwanatha Shetty, was stabbed in his office by a disturbed man for not getting a government order through a tender. Similarly wards of a number of Ministers are also in the fray promoting politics of family in the state.
What Robert Mueller Reportedly Wants To Ask Donald Trump
The New York Times published a set of questions meant for Trump reportedly from the office of the special counsel. Trump's lawyers have been negotiating the details of a potential interview between the president and Mueller.
Burning Skyscraper Collapsed in Sao Paulo
The BBC reports that the fire began around 1.20am local time (4.20am GMT), and that over 100 firefighters attended the scene. Highly organized squatters have taken over as many as 70 buildings in the downtown area of South America's largest city.
Today marks World Press Freedom Day
This comes days after India slipped two positions in the World Press Freedom Index 2018 to Rank 138 in theb list of 180 countries. Journalism is considered to be a risky profession as well, with more than 2,500 journalists having been killed since 1990.