"Labour will only vote for a final Brexit deal if it delivers a strong relationship with the single market based on full tariff-free access and ensures no loss of rights and standards", Labour leader Jeremy Corbyn said on Facebook.
Either way, the chances are increasing that Parliament will get a greater say over Brexit.
"The main reason for my taking this decision now is the Brexit process and the government's wish to limit parliament's role in contributing to the final outcome in a vote that takes place today", Lee, who voted to remain in the European Union during Britain's 2016 referendum, said on his website.
"First, we must never do anything that undermines the Government's negotiating position or encourages delays in the negotiations", Mr Davis said.
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. 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.
Theresa May's government gives in to Tory rebels in order to avoid a major defeat on the EU Withdrawal 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.
Just hours before the vote, the pressure on Ms.
"I absolutely trust what the Prime Minister says to us", he told BBC Radio 4's PM programme.
Opening the debate, Brexit Secretary David Davis insisted the government would abide by three principles to defend the will of the British people.
"The question of what form parliamentary approval of the withdrawal bill takes is one of the most significant decisions this house will have to take", he said.
If May is defeated by a wide margin her position as Prime Minister could be threatened.
During a frantic day of discussions between ministers and Conservative backbenchers, potential rebels were eventually persuaded to back down when Solicitor General Robert Buckland told MPs that ministers were willing to "engage positively" with their concerns.
Jacob Rees-Mogg, chair of the influential European Research Group of MPs, urged Mrs May to reject outright Mr Grieve's further call for MPs to effectively take control of negotiations in the last resort if no deal is agreed by February 2019.
The key Brexit bill is back in the Commons today and tomorrow for a series of knife-edge votes that will determine the future of the Brexit process. May's been resisting the demand because she doesn't want her hands to be tied during the talks.
They also voted to disagree with Lords amendment 37, which was part of an attempt to remove the exit day from the Bill and allow the Commons to rethink its approach.
Conservative lawmaker Phillip Lee, who voted in Britain's 2016 referendum to remain in the European Union, resigned as a justice minister so he could vote against the government on a measure that would give Parliament more power over the terms of the break.
A statement from the Brexit ministry said the government had agreed to "look for a compromise".