Davis and Prime Minister Theresa May have argued that they can not accept anything which gives parliament the power to bind their hands in negotiations with the European Union, or opens the door to lawmakers overturning the result of the 2016 Brexit referendum.
"Parliament must be given a meaningful vote on all possible outcomes of Brexit - and this must be guaranteed in writing".
He told Sky News the thought of that "wakes me up in the middle of the night in a cold sweat".
Grieve who had said "we could collapse the government" in an interview at the weekend, stressed he was talking about what might happen if the final Brexit withdrawal agreement were rejected - something which fellow backbencher Tom Tugendhat had also claimed.
The amendment would require the government to allow MPs to vote on how it should proceed if there is no Brexit deal by 21 January 2019.
He said: "One of the reasons I've supported [this amendment] is precisely to avoid a situation where the government would immediately collapse", he said.
Speaking before the vote, foreign minister Boris Johnson reinforced the government's view that discussion of the meaningful vote was hypothetical as ministers were confident of getting a deal with Brussels that parliament will approve.
Under these circumstances, the government has said, a minister will make a statement in Parliament, setting out the government's next steps.
"The whole point is to send to the Commons the agreement Dominic Grieve and others thought they had reached with the Prime Minister".
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She insisted she had been listening to the concerns of critics but said the legislation must not restrict her freedom in talks with Brussels.
If the United Kingdom looked like leaving the European Union without a deal, he said it would be a moment of "national crisis".
She refused to "tie the hands of government" in ongoing negotiations.
He added: "The Government's amendment not only fails to deliver the promised meaningful vote".
Crucially, the motion will be unamendable, meaning that MPs can not insert a requirement for Mrs May to go back to the negotiating table, extend the Brexit transition or revoke the UK's withdrawal under Article 50.
The minister leading those negotiations, Solicitor General Robert Buckland, said the government was standing by its own proposal, rejected by party rebels last week.
"I would say to all of my colleagues: vote in a way to make sure the Government can get the best deal for the United Kingdom, because that's what the people of the United Kingdom need and deserve".
"To rule out "no deal" as an option completely, even as a theoretical negotiating objective, places the government in an impossible position", former Chancellor of the Exchequer Norman Lamont told lawmakers.