Theresa May said both sides will have to compromise in cross-party Brexit talks with Labour as she faced a backlash over the negotiations with Jeremy Corbyn's party.
Further talks are expected this week and shadow business minister Rebecca Long-Bailey, a member of Labour's negotiating team, said while it was "disappointing" that there had not been any shift in the Government's red lines "the overall mood is quite a positive and hopeful one".
Labour's key demand is for a customs union with Brussels in order to protect the flow of goods, but Brexiteers vehemently oppose anything that would restrict the UK's ability to strike free trade deals through being bound by tariffs set by the EU.
In a video message filmed at her Chequers country retreat, Mrs May said: "There are lots of things on which I disagree with the Labour Party on policy issues.
"But on Brexit I think there are some things we agree on: ending free movement, ensuring we leave with a good deal, protecting jobs, protecting security.
"And so we are talking. Can we find a way through this that ensures that we can get a good deal and a deal agreed through Parliament?
"It'll mean compromise on both sides but I believe that delivering Brexit is the most important thing for us."
Ms Long-Bailey told BBC's Andrew Marr Show that in the talks Labour had discussed how any changes to the Brexit agreement "could be entrenched" so that any potential future Conservative leader, such as Boris Johnson, would not be able to "rip up" any compromise – a so-called "Boris-proof" deal.
She said a customs union was "defined in international law" and "the proposals we have seen from the Government so far and their direction of travel over the last two years have not been compliant with the definition of a customs union".
But Commons Leader Andrea Leadsom said the Prime Minister's deal with Brussels already had a "customs arrangement" aimed at tariff-free trade.
She added: "My expectation – and I'm not party to the discussions – is that the Prime Minister will only seek to agree those things that still constitute Brexit."
The Prime Minister heads to Brussels on Wednesday for an emergency summit aimed at securing a further delay to Brexit, with Mrs May hoping for an extension until June 30 at the latest, with the option of leaving the EU earlier if a deal can get through Parliament.
If no extension is agreed then the UK is set to leave without a deal on Friday.
"It is a very hypothetical question"
— BBC Politics (@BBCPolitics) April 7, 2019
Ms Long-Bailey suggested Labour could be prepared to revoke Article 50, cancelling Brexit, if the UK was heading towards a no-deal scenario.
"We have promised our party members and our constituents that we will do all we can to avoid a no-deal situation," she said.
Signalling that talks aimed at reaching a deal with the Tories were expected to continue she said "hopefully at the beginning of next week we will be having further discussions".
Ms Long-Bailey also said that Labour had raised the prospect of a second referendum to prevent a "damaging" or no-deal Brexit.
Some 80 MPs, including frontbenchers, have written to Jeremy Corbyn demanding a public vote on any deal that emerges from the talks.
Brexiteer Mrs Leadsom said she would never back the revocation of Article 50, insisting the UK could "survive and thrive" after a no-deal exit and it would be "not nearly as grim as many would advocate".
— BBC Politics (@BBCPolitics) 7 April 2019
The Commons Leader said the Tories were working with Labour "through gritted teeth", adding that a second referendum is an "appalling" idea and taking part in the European elections would be "utterly unacceptable".
Tory Eurosceptics are furious at the Prime Minister's handling of the Brexit process.
European Research Group chairman Jacob Rees-Mogg told Sky News: "The Prime Minister could have taken us out on March 29. It was the Prime Minister who asked for an extension, it was the Prime Minister who changed the date by prerogative power from March 29 to April 12.
"This all rests with her and upon her shoulders. The Prime Minister, Mrs May, has made active choices to stop us leaving and she deserves to be held to account for that."
Ex-whip Michael Fabricant predicted "open revolt" in the Conservative Party and among Leave voters if Mrs May agreed to a customs union.
The Sunday Telegraph reported that Conservative activists are refusing to campaign for the party and donations have "dried up" because of Mrs May's leadership.
In a letter to the Prime Minister, more than 100 current and would-be Tory councillors state that they are unable to muster the volunteers needed to effectively fight next month's local elections because "belief in the party they joined is gone".
Former Brexit secretary Dominic Raab said Mrs May's approach "threatens to damage the Conservatives for years" and that teaming up with Mr Corbyn could be "potentially disastrous for the nation".
Writing in the Mail on Sunday, he said: "There is now a danger that Brexit could be lost and that the Government could fall – handing the keys to Downing Street to Corbyn."