He said Mrs May promised to table amendments in the House of Lords that will address their concerns.
Tory former attorney general and leading pro-Europe rebel Dominic Grieve urged ministers to back his compromise amendment on the meaningful vote.
The government is resisting the changes, which are aimed at softening the terms of the U.K.'s exit from the European Union.
The government is trying to overturn 14 amendments from the House of Lords, after accepting one which May's spokesman said "essentially states government policy in relation to the fact that we could have future engagement with some European Union agencies".
His competing amendment could force ministers to hand over control of its Brexit strategy to parliament if there is no deal by mid-February.
The amendment, which would push the government towards adopting a Norway-style deal with Brussels, was voted down by 327 votes to 126, a majority of 201.
He said: "I understand the difficulties MPs representing constituencies which voted strongly for Leave or Remain have on the EEA amendment to the EU Withdrawal Bill".
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But part C - arguably the most controversial section of the proposal - has been pushed back, with Theresa May apparently promising to have further discussions in the Lords and Commons as ping pong between the two Houses of Parliament continues.
On Tuesday, parliament will also debate other amendments, including a challenge to the government's plan to put March 29, 2019, or "Brexit Day", into law and an attempt to toughen a commitment to ensure a frictionless border between Northern Ireland and the neighbouring Irish Republic, which will remain in the EU. They should find an agreement by the end of the week before the bill returns to the House of Lords.
Remainer Stephen Hammond said a group of potential rebels - believed to number 15 to 20 - received assurances from the PM moments before the key vote.
Speaking on BBC radio on Tuesday morning, Mr Davis said a meaningful vote would not lead to Britain staying in the EU.
There was little doubt the government would win on the customs union and single market, which some pro-EU lawmakers say is the only way for Britain to retain economically advantageous close ties with the bloc, with the opposition Labour Party also divided over future relations.
Writing in The Guardian, shadow Brexit secretary Sir Keir Starmer said: "Instead of negotiating for Britain, the Prime Minister is lurching from crisis to crisis, increasing the risk that the talks break down, and we crash out without an agreement".
"If, in the future, I am to look my children in the eye and honestly say that I did my best for them I can not, in all good conscience, support how our country's exit from the European Union looks set to be delivered".
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For that to happen though, it is important that the iPhone be allowed to cross the USA and China borders unfettered by tariffs. Last week he threatened 25 percent tariffs on $50 billion in Chinese products with another $100 billion targeted beyond that.
In the event, Dr Lee abstained on the crucial vote, saying he was "delighted" the Government had agreed to introduce an amendment giving Parliament "the voice I always wanted it to have in the Brexit process".
The Lords passed an amendment stating that Brexit can not go ahead until a minister has presented a report setting out what the United Kingdom has done to try to negotiate continued customs union membership.
It's an amendment that calls for the government to "take all necessary steps to implement an worldwide trade agreement which enables the U.K.to participate after exit day in a customs union with the EU in the same terms as existed before exit day".
Justice Minister Philip Lee resigns and pledges to vote against Theresa May on Brexit.
And the fact it's so softly worded might make timid rebels think they can vote for it without causing the prime minister too much harm.
Liberal Democrat Brexit spokesman Tom Brake said: "As has become a tradition in Brexit negotiations, the Tories have been forced to cobble together a compromise".
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