Prime Minister Theresa May has insisted she had to reach out to Labour in a bid to deliver Brexit or risk letting it "slip through our fingers".
The PM said there was a "stark choice" of either leaving the European Union with a deal or not leaving at all.
And shadow business secretary Rebecca Long-Bailey says if no-deal became an option Labour would consider "very, very strongly" voting to cancel Brexit.
Some Tories have criticised the PM for seeking Labour's help on her deal.
Commons Leader Andrea Leadsom said the Tories were working with Labour "through gritted teeth", adding that no deal would be better than cancelling Brexit.
MPs have rejected Mrs May's Brexit plan three times and last week's talks between the two parties were aimed at trying to find a proposal which could break the deadlock in the Commons before an emergency EU summit on Wednesday.
However, the three days of meetings stalled without agreement on Friday.
In a video message posted on Sunday, Mrs May said she could not see MPs accepting her deal "as things stand".
She added that she had been looking for "new ways" to get a deal through Parliament, but it would require "compromise on both sides".
"I think people voted to leave the EU, we have a duty as a Parliament to deliver that," she added.
Labour leader Jeremy Corbyn has said he was "waiting to see the red lines move" and had not "noticed any great change in the government's position".
He is coming under pressure from his MPs to demand a referendum on any deal he reaches with the government, with 80 signing a letter saying a public vote should be the "bottom line" in the negotiations.
'Willing to continue talks'
In a statement issued on Saturday night, Mrs May said after doing "everything in my power" to persuade her party - and its backers in Northern Ireland's DUP - to approve the deal she agreed with the EU last year, she "had to take a new approach".
"We have no choice but to reach out across the House of Commons," the PM said, insisting the two main parties agreed on the need to protect jobs and end free movement.
"The referendum was not fought along party lines and people I speak to on the doorstep tell me they expect their politicians to work together when the national interest demands it."
Getting a majority of MPs to back a Brexit deal was the only way for the UK to leave the EU, Mrs May said.
"The longer this takes, the greater the risk of the UK never leaving at all."
Ms Long-Bailey, who was involved in Labour's meetings with the government, told BBC's Andrew Marr Show they were "very good-natured" and there had been "subsequent exchanges".
She said Labour was yet to see the compromise proposals needed to agree a deal but she was "hopeful that will change in the coming days and we are willing to continue the talks".
However, she added Labour would "keep all options in play to keep no deal off the table", including supporting a vote to revoke Article 50 - the legal mechanism through which Brexit is taking place.
Tory Brexiteers have reacted angrily to the prospect of Mrs May accepting Labour's demands, particularly for a customs union with the EU which would allow tariff-free trade in goods with the bloc but limit the UK from striking its own deals.
Ms Long-Bailey indicated Labour might be willing to be flexible over its support for a customs union but said the government proposals on the issue have "not been compliant with the definition of a customs union".
Interviewed on the Andrew Marr Show, Ms Leadsom reiterated her comments in the Sunday Telegraph that holding another referendum on the UK's departure would be the "ultimate betrayal".
She said that taking part in the European elections in the event of a Brexit delay would be "utterly unacceptable".
Ms Leadsom said: "Specifically provided we are leaving the European Union then it is important that we compromise, that's what this is about and it is through gritted teeth. But nevertheless the most important thing is to actually leave the EU," she said.
The Commons leader also told the BBC's Brexitcast there is the potential for bringing Mrs May's deal back before MPs this week.
The UK is due to leave the EU on 12 April and, as yet, no withdrawal deal has been approved by the House of Commons.
This week Mrs May is to ask Brussels for an extension to 30 June, with the possibility of an earlier departure if a deal is agreed.
Labour says it has had no indication the government will agree to its demand for changes to the political declaration - the section of Mrs May's Brexit deal which outlines the basis for future UK-EU relations.
The document declares mutual ambitions in areas such as trade, regulations, security and fishing rights - but does not legally commit either party.
Leaving the EU's customs union was a Conservative manifesto commitment, and former party whip Michael Fabricant predicted "open revolt" among Tories and Leave voters if MPs agreed to it.
However, Downing Street has described the prospect as "speculation".
Meanwhile, the Sunday Telegraph reported some activists were refusing to campaign for the party, while donations had "dried up".
And former Brexit Secretary Dominic Raab writes in the Mail on Sunday that Mrs May's approach "threatens to damage the Conservatives for years".
"There is now a danger that Brexit could be lost and that the government could fall - handing the keys to Downing Street to Corbyn," he says.
Tory MP Jacob Rees-Mogg said including Mr Corbyn in the Brexit process was a "mistake" as "he is not sympathetic to the government, obviously, and is a Remainer".
He told Sky News the reason Mrs May has not been able to secure the backing of all Conservative MPs was "her own creation" and because she failed to "deliver" a deal they could support.
Treasury Chief Secretary Liz Truss dismissed the idea of a long delay to Brexit, which could be ended if Parliament approved a deal.
Ms Truss told BBC Radio 5 Live's Pienaar's Politics a so-called flextension "sounds like purgatory", adding: "We haven't yet negotiated the free trade deal we need... So I think the British public are going to be pretty horrified if we go into more limbo than we've already had."
In a letter to Mr Corbyn, some Labour MPs have pointed out that - because the political declaration is not legally binding, and with Mrs May having promised to stand down - a future Tory PM could simply "rip up" any of her commitments.
Four shadow ministers were among 80 signatories of the Love Socialism Hate Brexit campaign letter pressing for a further public vote.
Any compromise deal agreed by Parliament will have "no legitimacy if it is not confirmed by the public", it argues.
However, Labour is split on the subject, with a letter signed by 25 Labour MPs on Thursday arguing the opposite.
They warned it would "divide the country further and add uncertainty for business" and could be "exploited by the far-right, damage the trust of many core Labour voters and reduce our chances of winning a general election".
Key dates in the week ahead
- Monday: Possible resumption of talks between the government and Labour
- Wednesday: Emergency summit of EU leaders to consider UK request for further extension until 30 June, with the option of an earlier Brexit day if a deal can be agreed
- Friday: Brexit day, if UK is not granted a further delay