Theresa May has written to all 317 Tory MPs, urging them to unite behind a Brexit deal while warning them “history will judge us all” over the process.
Efforts will resume on Monday to persuade the EU to agree changes to the “backstop” plan to prevent the return of customs checks on the Irish border.
And Culture Secretary Jeremy Wright has hinted MPs’ concerns about it could be addressed without reopening the deal.
Labour says the Tories cannot be united and has called for cross-party talks.
The UK remains on course to leave the EU on 29 March. But Mrs May has been unable to convince a majority of MPs to back the withdrawal terms she struck with the EU last year.
The prime minister told MPs in the letter she will return to Brussels to meet European Commission President Jean-Claude Juncker this week, and speak to the leaders of every EU member state over the coming days.
Her main goal is to win concessions over the backstop, which is widely disliked by members of her party.
Many fear it will mean the UK staying closely aligned to EU rules for the long term, without Britain being able to end the agreement unilaterally.
But EU leaders have repeatedly said the withdrawal agreement is not open for renegotiation.
The Sunday Times reported comments it said were leaked from a WhatsApp group suggesting ex-Brexit minister Steve Baker told fellow Brexiteers that Mrs May’s talks with Brussels were a “complete waste of time”.
However, Culture Secretary Mr Wright has hinted that there might be “a number of different ways” around the problem.
“I don’t think it’s the mechanism that matters, it’s the objective,” he told the BBC’s Andrew Marr Show, when asked whether a codicil – a supplementary document explaining or modifying a legal agreement – might work.
“Parliament needs to give the prime minister space to have that conversation with Brussels,” he added.
Attorney General Geoffrey Cox, who has warned the backstop indefinitely commits the UK to EU customs rules if Brexit trade talks break down, will set out what changes would be needed to address concerns in a speech on Tuesday.
Brexit Secretary Stephen Barclay is also due to meet the EU’s chief negotiator Michel Barnier to discuss the controversial policy on Monday.
If MPs do not approve a formal deal, many fear chaos at ports and for business.
And Tobias Ellwood has become the first minister to publicly declare a willingness to rebel against the government if the PM failed to rule out a no-deal scenario.
“There are many ministers, me being one of them, that need to see ‘no deal’ removed from the table,” the defence minister told BBC Radio 5 Live’s Pienaar’s Politics.
Labour’s shadow chancellor John McDonnell told Andrew Marr that “serious discussion” around building cross-party consensus must happen by the end of the month.
“We have put our proposals on the table; we’re willing to negotiate around those,” he said. “They’re the basis of what we think could secure parliamentary approval, but she has to start negotiating as well.”
The PM’s negotiating stance has never been “based upon mutual interest or mutual respect” but was “about banging the table and walking away”, he added.
Asked about the prospect of a further referendum, Mr McDonnell said the party’s priority remained a Brexit deal that protected jobs and the economy.
But he added: “We really are at the end of the line now… If [a deal] doesn’t fly within Parliament, yes the option of going back to the people has got to be there.”
Campaigners for another referendum on Brexit have said there will be a major protest the weekend before Britain’s scheduled departure date on 29 March.
‘Failure to make compromises’
In her letter, Mrs May described the latest Commons defeat over her Brexit strategy, in which dozens of Conservatives abstained on Thursday, as “disappointing”.
“I do not underestimate how deeply or how sincerely colleagues hold the views which they do on this important issue – or that we are all motivated by a common desire to do what is best for our country, even if we disagree on the means of doing so,” she writes.
“But I believe that a failure to make the compromises necessary to reach and take through Parliament a withdrawal agreement which delivers on the result of the referendum will let down the people who sent us to represent them and risk the bright future that they all deserve.”
‘No change in tactics’
Analysis by Peter Saull, BBC political reporter
Since Thursday and the 10th defeat in the House of Commons for Theresa May over Brexit, the tensions in the Conservative Party have threatened to boil over.
The war of words has reached ministerial level, with business minister Richard Harrington suggesting some of his pro-Brexit colleagues should join former UKIP leader Nigel Farage’s new party.
Therefore while there’s nothing new in terms of the substance of this letter, it represents a clear attempt by the prime minister to calm things down.
Theresa May is also making it clear she won’t change tactics.
The PM believes the only way to get a deal through the Commons and keep her party together is by securing changes to the backstop, even if the EU shows no sign of budging.