Theresa May has urged her party to support a demand for the backstop to be replaced with "alternative arrangements".
The UK prime minister has asked Tory MPs to back the position during a series of Brexit votes in the House of Commons on Tuesday.
This proposal is being brought by Conservative backbencher Sir Graham Brady despite Brussels insisting that there will be no alteration on the backstop - which is the guarantee of no hard border in Ireland.
Conservative chairman Brandon Lewis announced that Tory MPs will be whipped to support an amendment tabled by senior backbencher Sir Graham in the series of votes on Tuesday evening.
And Tory backbenchers Andrew Murrison and John Baron withdrew their rival amendments, which would have required the scrapping or time-limiting of the backstop, which is designed to avoid the need for a hard border in Ireland following Brexit.
But the dramatic move was far from guaranteed to deliver a majority for the plan, after it failed to win the support of the influential European Research Group of Conservative eurosceptics.
However Cabinet minister Damian Hinds said alternatives to the controversial backstop were already being considered.
Asked if the Government would support the Brexit amendment backed by Sir Graham, the Education Secretary told BBC Radio 4's 'Today' programme: "The Prime Minister has been clear already that alternative arrangements are an alternative to the backstop - as by the way, is extending the implementation period.
"So it is not necessarily the case that if we did reach that point the backstop would have to come in anyway.
"It is an uncomfortable thing for the European Union, as it is an uncomfortable thing for us - there are some things about it, like the fact that we would have free market access but without paying, free movement would be over, we would be out of the common agricultural policy and fisheries policy - that's not comfortable for the EU either."
Sir Graham said his amendment would give "enormous firepower" to Theresa May when she returns to Brussels by setting out what the UK Parliament would back.
He told 'Today' that any alternative arrangement would have to be "legally binding" as "I don't think anybody is going to accept something which is just warm words".
Sir Graham said his amendment was drawn up after meetings with ministers including Mrs May.
Asked if it had Government backing he said: "I hope so... I don't know so. The amendment was born out of a number of conversations I had with colleagues including members of the Government, including the Prime Minister.
"I had also spoken to people in the DUP too and I'm hoping that the way in which the amendment is crafted can attract that very broad support and if we can win the vote on my amendment then I think it gives the Prime Minister enormous firepower."
But European Commission spokesman Margaritis Schinas again insisted that Brussels would not reopen the Withdrawal Agreement negotiations.
"We have a unanimous EU27 position on the Withdrawal Agreement which reflects the common EU position," he said.
"This Withdrawal Agreement has been agreed with the UK Government, it is endorsed by leaders and is not open for renegotiation."
Asked if that position would change if MPs vote to demand changes to the Irish backstop, Mr Schinas said: "The only thing I have to say is that we shall wait for the result of the vote of the Commons tomorrow.
"Then we will wait for the Government to tell us what are the next steps. That's how it's going to work."
Brussels' senior officials will consider the latest "Brexit preparedness and contingency" measures at this week's College of Commissioners meeting on Wednesday.
Tuesday's House of Commons debate on the Government's EU withdrawal deal will be opened by Brexit Secretary Stephen Barclay and closed by the Prime Minister.
The day's debate will be followed "as soon as possible" by a second meaningful vote on whatever deal has been secured with Brussels, said the spokesman. Like the meaningful vote defeated by a margin of 230 earlier this month, this will also be amendable.
It is understood that the second meaningful vote - expected to take place in February - will go ahead whether or not the EU has agreed to amend the Withdrawal Agreement reached with the UK in November.
Mrs May's spokesman was asked whether there was any indication that Brussels might be ready to reconsider the backstop arrangements, either by reopening the Withdrawal Agreement or by attaching further conditions or assurances in the form of a codicil.
He replied: "I think there is recognition that the UK leaving with a deal is in their interests as well as those of the UK.
"If we are going to leave with a deal, clearly we are going to have to make some changes to secure parliamentary support."
The spokesman added: "I don't envisage any circumstances in which there would not be a second meaningful vote."
Meanwhile Sammy Wilson, the DUP's Brexit spokesman, has urged Theresa May to "face down the stubbornness" of Dublin and Brussels over her Withdrawal Agreement.
He said the Prime Minister should "exploit the cracks" emerging in their "illogical" position as he urged Ireland to work "constructively" with the UK.
Mr Wilson added: "Whether it was waving front page stories about bombs 30 years ago or silly talk about troops being sent to the border, the Irish government has exploited the Brexit negotiations for its own narrow political domestic aims. To use border communities in such a way is quite disgraceful.
"Leo Varadkar may have thought in his naivety that it was wise to suck up to the EU. Maybe he even hopes to get a post in Europe in the future, but in reality he should be cautious."