"Further amendments to the Brexit bill will be discussed today; but even after that I still do not see a clear path that will satisfy both sides of the party, the Brexiteers and the Remainers, sufficiently so as not to topple the government and that will also seem acceptable to the European Union", says Esther Maria Reichelt, an analyst at Commerzbank.
"First, we must never do anything that undermines the Government's negotiating position or encourages delays in the negotiations", Mr Davis said.
Conservative lawmaker Phillip Lee, who had voted to remain in the European Union, resigned as the justice minister so he could speak out against the policy on Brexit.
Setting out the approach for a customs union and "full access to the internal market" which would be "underpinned by shared institutions and shared regulations", he said: "That's the type of deal either this government will be forced to enter into or a Labour government will implement".
Another flashpoint could come when lawmakers vote Wednesday on an amendment seeking to keep Britain in a customs union with the EU.
It was a second win for May after she persuaded rebels in her Conservative party on Tuesday to reject a Lords amendment that would have allowed parliament to block the government from leaving the European Union with no deal. "There are choices that we face, there are trade- offs that we have to accept". However, the party is divided on the issue, with a total of 89 Labour MPs choosing to defy Corbyn. It meant that the United Kingdom government successfully defeated the Lords amendment, 324 to 298, to the dismay of Labour and other opposition parties.
What do the newspapers say about the votes?
The most contentious was the bid to give Parliament the power to tell the government what to do if the Brexit deal was voted down or no agreement was reached.
The Sun had a frontpage of British icons including Stonehenge, a fish and chip shop, a London bus and a football, saying "Great Britain or Great Betrayal". The Daily Express featured the British flag as its front page with the headline: "Ignore the will of the people at your peril". It will then be discussed in the upper chamber of parliament on June 18, before returning to the elected lower house for approval at an as-yet unspecified date in the near future.
But government officials admitted that the concession does constrain their freedom in the Brexit talks.
The land border, which separates European Union member state Ireland from the rest of the United Kingdom, now has no checks on people or goods crossing from side to side. The formal separation is to happen in March 2019. An open border is one of the key guarantees of the Good Friday Agreement, which ended decades of sectarian violence in Northern Ireland.
That said, while the government may have won the battle on customs, the war is still to come. "I am trying to negotiate the best deal for Britain".
It is a politically explosive issue with no obvious solution.
Yvette Cooper, the former shadow home secretary who now chairs the home affairs select committee said she would obey the party whip and not vote for membership of the EEA only because she wanted to try to build consensus.
But she told MPs that "the government's hand in negotiations can not be tied by parliament".