The disagreement centers on whether the government agreed to consider a specific clause of the rebel proposal that would hand parliament control of the Brexit process if ministers are unable to strike an exit deal by February 15, 2019.
Created to address concerns aired by pro-EU Conservative backbenchers, the amendment suggests that Parliament should have a say on the final Brexit deal.
Under these circumstances, a minister must make a statement in Parliament within 14 days and give MPs an opportunity to vote.
He said: "I think it is unacceptable because it seems to me to be contrary to what the whole intention was behind this whole amendment".
"I voted on Tuesday to make sure that the sovereign institution of this nation, our Parliament, gets its full sovereign right to review our new relationship with the trading partners we will have in future".
Mr Grieve said, despite being aware of the possible catastrophic consequences of his actions, he would pursue a vote against the Government's Brexit bill next week if concessions are not made before then.
Under the Government's plan, MPs will only be able to debate the Prime Minister's next move if no deal is struck or they vote down the deal, and would have no power to direct what should happen next.
He added: "But this remains hypothetical and the government is confident we will agree a good deal with the European Union which Parliament will support".
British Prime Minister Theresa May said Sunday it is important to ensure that Parliament cannot block Brexit, but one of the leading MPs in her governing Conservative Party warned that her government could collapse if she does not allow Tory dissidents more influence over the decision and its final terms.
"Grateful for the conversations but without consultation what was agreed earlier today has been changed", she said.
But the latest turn of events raises the stakes and leaves Mrs May's premiership in jeopardy if she loses the vote on the amendment in the Commons on Wednesday.
Tory Remainer Sarah Wollaston tweeted: 'So just to be clear we are now going to have to amend the "unamendable" after the agreed amendable amendment acquired a sneaky sting in the tail.
"The Government's amendment is simply not good enough", the Shadow Brexit Secretary Keir Starmer said. "Parliament can not - and should not - accept it".
"This meaningful vote must be comprehensive and offer the possibility of MPs putting the deal to the people". They can not be trusted with Brexit.
She hailed Grieve as a "hero" for supporting a final say on Brexit.
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.