Theresa May is being urged to secure changes from the EU to the Northern Irish backstop as part of her Brexit deal to get it past parliament.
Ex-foreign secretary Boris Johnson says winning a “freedom clause” would be “unadulterated good Brexit news”.
MPs on Tuesday will vote on a series of amendments to the PM’s plans that could shape the future direction of Brexit.
But the Irish deputy PM says changes to the backstop – aimed at preventing a hard border – would not be acceptable.
The backstop plan – a fallback position to avoid the return of customs checks at the Irish border if the UK fails to agree a deal with the EU – is a major sticking point for opponents of Mrs May’s withdrawal agreement.
It has proved unpopular with many Brexit-supporting Conservative MPs who are concerned that it could tie the UK to many EU rules indefinitely.
After Mrs May’s deal was rejected in the Commons on 15 January by 432 votes to 202, opposition and backbench MPs have been tabling amendments to the plans.
While MPs will not deliver their final verdict on the deal on Tuesday, they will vote on the amendments.
One that has the backing of senior Tory Sir Graham Brady, the chairman of the backbench 1922 Committee, calls for the backstop to be replaced by “alternative arrangements”.
Speaking on BBC Radio 4’s Westminster Hour, Sir Graham said it was the “kind of compromise [way] forward that can actually attract the support of a majority in the House of Commons. I hope it will and demonstrate in terms to the EU there is a way forward for the withdrawal agreement.”
Brexiteers say if the non-binding amendment is passed it will give the PM more leeway to win concessions from the EU.
Writing in the Daily Telegraph, Mr Johnson suggested Mrs May could get the EU to change the text of her agreement to include an expiry date for the backstop, or include a mechanism allowing the UK to unilaterally withdraw from it.
Mr Johnson said: “If the PM secures that change – a proper UK-sized perforation in the fabric of the backstop itself – I have no doubt that she will have the whole country full-throatedly behind her.”
One of the architects of the Good Friday Agreement, Lord Bew, told BBC Radio 4’s Today programme that the backstop must be made “explicitly temporary and in legally binding terms” in order to avoid undermining the Agreement.
He said: “There is a conflict, the backstop advertises itself as defending the Good Friday Agreement in all its aspects.
“But you only have to look at what it says about agriculture… and you will see that you have a totally different top-down approach to dealing with matters of agriculture concern north and south to that in the Good Friday Agreement, where incidentally it’s the very top item for co-operation.”
Reiterating earlier comments that the EU would not ratify a deal without the backstop being included, the Irish Republic’s deputy prime minister and foreign minister Simon Coveney said: “Even in a no-deal Brexit situation every party and every MP in the UK will have a responsibility to ensure there is no return to a hard border and Northern Ireland is protected.”
He added: “The EU has been clear that the backstop is an integral part of the withdrawal agreement.”
The UK is due to leave the EU at 23:00 GMT on 29 March, and the prime minister has faced repeated calls to rule out the prospect of leaving without a deal if her agreement does not win approval.
It is also possible that MPs will back amendments on Tuesday which call on Mrs May to rule out no deal or to delay the UK’s departure from the EU.
An amendment tabled by Labour MP Yvette Cooper could extend Article 50 – which triggers the UK’s withdrawal from the EU – by nine months, unless the prime minister can secure a deal by the end of February.
Appearing on the BBC’s Andrew Marr Show on Sunday, Ms Cooper said she was not seeking to “block Brexit”, adding that the bill would be amendable – meaning MPs could vote on how long any extension would be.
Meanwhile, a report from MPs on the Exiting the EU Committee says the government must rule out a no-deal Brexit.
Committee chairman Hilary Benn said: “The suggestion that the UK might opt for a no-deal outcome but assume that the EU will continue to act in a co-operative manner to avoid disruption, cannot seriously constitute the policy of any responsible government.”
However, a Tory member of the committee, Craig Mackinlay, said he “disowned” the report findings as “just more Project Fear from a group of MPs who have never wanted the UK to leave the EU”.