But Labour MPs will now be told to vote for it as the Commons considers a series of options for the next phase of the Brexit process.
The EU has so far been unwilling to consider reopening the withdrawal agreement it reached with May, and which was subsequently rejected by parliament in a January 16 vote.
Former MI5 chief Baroness Manningham-Buller warned that a no-deal Brexit should be "avoided at all costs".
The plan is said to have been relatively warmly received by Conservative MPs across the Brexit divide.
This week will be crucial for Brexit as Prime Minister Theresa May sets out how she intends to take the departure process forward after her initial withdrawal agreement with the European Union was defeated.
The amendments are not legally binding on the PM, but majority support for any would place enormous political pressure on Mrs May as she seeks a way ahead after the crushing defeat of her plan by 230 votes earlier this month.
Meanwhile the PM is scrambling to defeat a motion backed by anti-Brexit campaigners would see Britain STAY in the European Union rather than quit without a deal.
People in the United Kingdom could face significant disruption to the supply of food if Theresa May fails to strike a Brexit deal with Brussels, leading retailers have warned.
Theresa May has thrown Government support behind an amendment that seeks to replace the controversial backstop with unspecified "alternative arrangements" to avoid a hard border in Ireland. Firstly, Mrs May would go back to Brussels and try to renegotiate the backstop element of her withdrawal agreement.
At this point, all May is trying to do is secure a majority so that she can get a win on the Brady Amendment.
Speaking to BBC News, Sir Graham said he hoped the backing from the government would see the DUP - who Mrs.
Former Conservative leader Iain Duncan Smith was reported to have called on Mrs May to table her own amendment, to provide greater "clarity and purpose".
European Union and Irish leaders insist that Dublin is under no pressure to relent.
Sir Graham Brady, chair of the influential 1922 Committee (responsible for hiring and firing Tory leaders) has tabled a Commons motion which would scrap the contentious Irish backstop, in a bid to get a Brexit deal through Parliament.
"I think this combines the two things, one that we will leave on the 29 whatever but it also ensures to those who are anxious about cliff-edges and not having any agreement that that departure will be a managed process should we not have arrived at an agreement beforehand - which the prime minister is now going back to renegotiate".
She characterised the proposals as "the withdrawal agreement we have got at the moment with changes to the backstop".
The petition states: "The British people MUST be given the Brexit they voted for anything else is not acceptable to the British public ARTICLE 50 must not under any circumstances be hindered/suspended/stopped for any reason whatsoever".
Ms Weyand told a Brussels conference that negotiation on the Withdrawal Agreement was "finished".
Mr Boles told the Today programme the plan has "a great deal of support among ministers and indeed Cabinet ministers" and they were pressing for a free vote.