Prime Minister Theresa May's Brexit plans face rejection by parliament's upper chamber on Monday, setting the stage for a confrontation with rebel lawmakers later in the week which could rock her minority government.
The battle over Parliament's say in the Brexit process between MPs, peers and ministers resumes in the Lords later.
In response to the change, 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.
At nearly the final moment on Tuesday, the two sides agreed to find a compromise and have until 1600 GMT to present it.
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.
On Tuesday MPs in the Commons voted to reject the amendment to the bill, which would weaken the PM's hand at the negotiating table. It is not a solicitation to make any exchange in commodities, securities or other financial instruments.
In a day of drama, May's position seemed suddenly weaker when junior justice minister Phillip Lee, who has always been critical of Brexit strategy, resigned and said he would vote against the government. Calls for MPS to have a "meaningful vote" on Brexit are nothing to do with parliamentary scrutiny; they are about stopping Brexit.
A spokeswoman for the Department for Exiting the European Union said the government's new amendment respected the tests set out by Brexit Secretary David Davis later on Tuesday evening, which state any compromise amendment must not undermine negotiations with Brussels or hinder the government's ability to negotiate worldwide treaties in future.
Brexit minister David Davis had earlier warned lawmakers that the government would never allow them to "reverse Brexit" or undermine negotiations.
"I think we are going to get a meaningful vote anyway". He added he would listen to the government but "I hope they listen to me when I say I don't understand why you've done this last-minute switch".
The rebels are seeking both closer ties with the European Union on issues like trade and customs, and to ensure that the government does not follow through on its threat to leave the bloc without a deal if it does not like the terms on offer from Brussels.
"This meaningful vote must be comprehensive and offer the possibility of MPs putting the deal to the people".
"It's also important that Parliament can not and should not overturn the will of the British people, which was to leave the European Union".
Labour's shadow Brexit secretary Sir Keir Starmer told the government their amendment is "simply not good enough". Instead, MPs will be offered a vote on an unamendable statement, which would effectively turn any vote into to a vote of confidence in the prime minister.