Davis and 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.
Theresa May's flagship Brexit legislation has once more been thrown into doubt, after a compromise created to keep Tory backbenchers on board was branded "unacceptable" by leading rebels.
The UK prime minister risks defeat on her Brexit legislation as she dares pro-EU rebels to vote her down.
Mr Grieve's amendment would have given Parliament the power to dictate the next steps if the Government failed to achieve a Brexit deal by February 2019.
"It is unacceptable", leading rebel Dominic Grieve told the BBC.
The rebels, led by former minister Grieve, stepped back from defeating the government on Tuesday after they were privately told that a concession on a no-deal veto would be forthcoming.
Mr Grieve indicated that the final text of the amendment tabled in the House of Lords at the last possible moment on Thursday had been changed from the wording which he believed had been agreed earlier in the day.
Tory rebels say a government amendment drawn up to avert a rebellion over the bill next week is not acceptable.
Next week the bill will return for debate in the House of Lords, with both the government amendment and Mr.
U.K. Prime Minister Theresa May appeared to have avoided defeat over the matter in parliament on Tuesday after giving assurances to rebels in her own party that she would make concessions in a new, government-backed amendment to the European Union (Withdrawal) Bill if they gave her their support.
"We need to recognize the role of Parliament, but ensure that the government's hands can't be tied in negotiations and that Parliament does not overturn the will of the people", May told the BBC.
Yet Brexiteers, including Jacob Rees-Mogg put pressure on the Government not to agree to the main thrust of the Remainers argument that Parliament should be able to instruct the Government how to negotiate with Brussels.
Shortly before the text of the amendment was tabled, former minister Anna Soubry tweeted to say that "deal or no deal Parliament will have a meaningful vote and ... there will be no hard Brexit".
"I'm endeavoring during the course of this afternoon to finalize agreement with the government concerning matters we debated yesterday", Grieve said in the House of Commons on Wednesday. At the end of the process something was inexplicably changed, which had not been agreed.
"If the government can't get the most important treaty through Parliament we'll be looking for a new government", said Mr Tugendhat.
"Would be amusing if only it wasn't such a serious issue".
Shadow Brexit secretary Sir Keir Starmer said: "Theresa May has gone back on her word and offered an amendment that takes the meaning out of the meaningful vote".
As with David Davis' constant resignation threats, there is a risk the rebels will turn into the boy who cried wolf - they have threatened again and again to make life hard for her, before capitulating on every vote.
The move means that Theresa May now faces the prospect of having her proposals discarded by the House of Lords next week. "Parliament can not - and should not - accept it".
But Labour MP Tom Tugendhat said the issue of the "meaningful vote" was secondary as the Government would fall if it could not secure parliamentary approval for its Brexit deal.