Britain warned the Brussels bloc it was not prepared to give up our rights on sovereignty, laws and fishing after Mr Frost and Mr Barnier shared a poignant moment. Mr Barnier tweeted that he and Mr Frost paused together in front of a war memorial following dinner on Monday in Westminster.
He said: "At the occasion of the working dinner with my counterpart David Frost at Carlton Gardens, a brief moment remembering our shared European history."
There continue to be a number of key sticking points in the way of agreeing a post-Brexit deal before the transition period ends in January.
The points of dispute - the "level playing field" of measures designed to ensure fair competition between the UK and EU, fisheries and the governance of any deal - will be on the agenda in Tuesday's sessions.
Ahead of the talks, Downing Street acknowledged significant differences remained between the two sides and reminded the EU it is "not asking for a special, bespoke or unique deal".
The Prime Minister's official spokesman said: In terms of the discussions that we have been having throughout the intensified process, they have continued to be constructive but significant differences still remain on a number of important issues.
Our position on our sovereignty, laws and fisheries is clear - we will not give up our rights as an independent state.
"We will continue to engage constructively with the EU on these key issues and will work hard to reach the broad outline of an agreement.
"But as we have been clear all along, we are not asking for a special, bespoke or unique deal."
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READ MORE:EU blasted for absurd and contradictory Brexit demands
5.31pm update:Brexiteers laud UKs exit as EU strides towards country called Europe
The European Union took another huge step towards a full-blown superstate on Tuesday after five days of bitter wrangling over a coronavirus rescue fund ended in a deal.
After a record-breaking summit negotiation marred by controversy, EU leaders finally agreed on a 750 billion spending plan for the blocs regions and industries worst hit by the pandemic. To pay for the aid, the European Commission was granted unprecedented borrowing, taxation and spending powers to run up 390 billion in joint debt to be handed out as grants rather than loans.
Conservative MPs said the pact showed Britain had quit the EU in the nick of time.
Former Brexit minister David Jones said: Had we still been a member, wed end up taking the second-largest share of bailing our countries right across the continent.
Fortunately were not and I think in hindsight more or less everybody, whether they voted to leave or not, will realise Britain has had a lucky escape on this occasion.
This is another step towards an increase in powers for the European Commission and reduction in power for the national government. Its a further step along the road to a creation of a country called Europe.
4.22pm update:Peers clash over post-Brexit farm support system
Peers have clashed over a new support system for farmers as the UK moves away from Europe's Common Agricultural Policy.
Liberal Democrat Lord Teverson called for a cut in the transition period to payments under new environmental land management schemes (ELMs).
But other peers warned, in committee stage debate on the Agriculture Bill, that a shorter transition period could hit farm incomes.
Lord Teverson told the Lords said it was better to introduce the new ELMs in five years rather than the seven proposed in the Bill.
But Tory Baroness McIntosh of Pickering said the transition period from direct payments should remain at seven years.
Lady McIntosh said it was the biggest change faced by farmers for a generation and a delay would give the Government time to develop a good scheme, rather than a "half-baked" one.
3.12pm update:Merkel accused of interfering in 2017 general election over Brexit
Angela Merkels Germany was accused of trying to influence the 2017 general election by undermining former Prime Minister Theresa May over Brexit talks, unearthed reports reveal.
According to a 2017 report by The Telegraph, senior officials in the German government and in Brussels openly mocked Theresa May in what was described as an attempt to undermine the former Prime Minister.
Ahead of the general election in June 2017, Mrs Merkel claimed Britain had illusions over what it could hope to achieve from Brexit.
Weakening Mrs Mays mandate with the electorate tilted the balance in favour of the EU negotiators, and Conservative sources suggested she was the victim of a coordinated plot.
Emily Ferguson is taking over live reporting fromRebecca Perring
2.25pm update: PM is confident Brexit was fair
British Prime Minister Boris Johnson is confident the 2016 Brexit referendum result was fair, his spokesman said on Tuesday.
Asked in the light of a parliamentary report into Russian influence in Britain whether Johnson thought the outcome of the referendum was fair, the spokesman said: "Yes, absolutely."
DON'T MISSFarage takes on EU again as Italexit campaign praises him [COMMENT]Italexit campaign No Europe - For Italy launches on Thursday [INFO]EU poised to stop trade with UK when the Brexit transition period ends[REVEALED]
1.56am update: Pompeo hails 'constructive' Brexit talks with Boris
Mike Pompeo said it was a constructive visit after he met Prime Minister Boris Johnson.
He said: "Our two countries' long-standing, strong bilateral relationship has laid the foundation for today's candid discussion on issues ranging from 5G telecommunication to our negotiations for a US-UK free trade agreement.
1.15pm update:Farage takes on EU again as Italexit campaign praises him for freeing UK from 'EU cage'
Nigel Farage is making his mark yet again in his battle against the EU after it emerged he has helped inspire the man behind Italys Italexit campaign.
Prominent Brexiteer and Brexit Party leader Nigel Farage has been hailed a true British patriot who sent away the technocrats from Brussels by anti-EU campaigner Gianluigi Paragone, a former senator for the anti-establishment 5Star Movement in Italy.
Mr Paragone revealed he had met with Mr Farage just before launching his Italexit campaign 'No Europe for Italy, which will take place on Thursday, July 23.
12.44pm update: Brexit will split financial markets - BoE
Brexit will make markets less efficient but it won't be disastrous for Britain's economy, an appointee to the Bank of England's Financial Policy Committee (FPC) said on Monday.
Jonathan Hall told the Treasury Select Committee that Brexit represented a longer term risk of increased fragmentation and complexity in financial services.
He said: "This would increase friction costs for the economy, the supervisory burden."
11.49am update: 'Brexit was always gift to Putin'
Guy Verhofstadt, chief Brexit negotiator for the European Parliament, tweeted following the release of the Russia report: "Brexit was always a gift to Putin because it weakened the European Union & left Britain divided, isolated. The #RussiaReport shows just how many questions remain unanswered."
11.25am update: Italy could follow in UK's Brexit footsteps
Italy will officially launch a campaign to unshackle itself from the EU on Thursday as the Mediterranean country looks to follow in Britain's Brexit footsteps.
The EU has another crisis on its hands after Italian politician Gianluigi Paragone, a former senator for the anti-establishment 5Star Movement, said it was time for Italy to launch its Italexit campaign ' No Europe for Italy' on Thursday, July 23. He said Italy would no longer be "blackmailed" by the Brussels bloc.
10.32am update: Russia meddled in Scottish referendum but unclear on Brexit - report
Russiameddled in the 2014 Scottish referendum and the British government failed to ask for a deep assessment of possible Kremlin-directed interference in theBrexitvote, the British parliament's intelligence and security committee said.
The report said: "There has been credible open source commentary suggesting thatRussiaundertook influence campaigns in relation to the Scottish independence referendum in 2014"
It said there were open source indications thatRussiasought to influence theBrexitcampaign but that the British government had not sought deep evidence of meddling.
The report castRussiaas a hostile power which posed a significant threat to the United Kingdom and the West across a range of fronts, from espionage and cyber to election meddling and laundering dirty money.
It added: "It appears thatRussiaconsiders the UK one of its top Western intelligence targets."
9.46am update: Pound Sterling enjoys rise
The pound briefly rose above $1.27 for the first time in six weeks on Tuesday.
The currency had enjoyed its best day in three-weeks on Monday, lifted by the generally buoyant market mood which allowed investors to overlook poor British economic data and lack of concrete progress on Brexit trade talks.
9.35am update:Nigel Farage lashes out at Remainers over Russia report
Nigel Farage has demanded serious apologies after an investigation reportedly found that Russia did not attempt to meddle in the Brexit vote.
The Brexiteer hit out after a long-awaited probe into alleged Russian interference in British democracy, due to be published later today, found nothing to suggest Russia played any part in the 2016 EU referendum, the Daily Telegraph reported.
Mr Farage tweeted: So, no evidence of Russian interference in the referendum. Some serious apologies are due.
8.48am update: US Mike Pompeo to meet Boris
As Britain toughens its stance on China due to its handling of the coronavirus and a crackdown in Hong Kong, U.S. Secretary of State Mike Pompeo's visit is an attempt to stiffen Boris Johnson's resolve and dangle the potential reward of a post-Brexitfree trade deal, diplomats say.
Mr Pompeo will discuss ways to tackle the growing might of China when he meets Mr Johnson on Tuesday, just a week after London ordered a purge of Huawei gear from the 5G network.
President Donald Trump lauded Mr Johnson's ban on Huawei, though he also claimed he had forced London's hand due to concern over China, which he considers to be the US' main geopolitical rival of the 21st century.
8.15am update: Policing minister says he doesn't know if Russia meddled in Brexit
Policing Minister Kit Malthouse said he does not know if Russia meddled in the Brexit vote or the Scottish independence referendum.
Asked on BBC Breakfast why it has taken so long for the report to be published, he said there has been a general election, adding: "And the fact that this report is the property of the committee itself and they make a decision about its publication.
"The fact that the committee took a little time to get going after the general election probably delayed it too.
"But it'll be here in, you know, a couple of hours, and we'll be able to read it and digest it."
Asked if he thinks Russia tried to influence the Scottish and EU referendums, he said: "I don't know. I haven't read the report. I'll have to wait and see what they have to say."
8.02am update:Taoiseach welcomes Brexit EU fund
Micheal Martinwelcomed a 5 billion reserve fund to support countries worst affected by the impact of Brexit.
It comes after the EU agreed on a 1.82 trillion budget and coronavirus recovery fund.
Micheal Martin said it is a strong deal which includes a substantial and significant package of measures.
The budget and recovery fund was finally agreed in the early hours of Tuesday after a marathon four-day summit in Brussels.
7.50am update: EU fishermen demand SAME access to UK waters in Brexit deal as system working well
Brexit trade talks with the EU will resume on Tuesday, but a breakthrough is not expected as the bloc is still adamant European fishermen must have the same access to UK waters - something British negotiators are keen to avoid.
Gerard van Balsfoort, Chairman of the European Fisheries Alliance, has told Michel Barnier to insist fishing rights remain the same after the conclusion of the Brexit transition period this year.
The EU's chief negotiator was told losing access to the waters could be disastrous for many EU states.
Mr van Balsfoort told the Telegraph: Loss of access to fishing grounds, to markets for fish or the return of overfishing will ultimately harm all of us.
"Michel Barnier knows this."
7.39am update: Brexit deal is possible but ambitious - Germany
German Foreign Minister Heiko Maas said on Tuesday: "The aim of reaching an agreement between Britain and the European Union on future ties between the two by October is ambitious but achievable.
An agreement on the basis of the Political Declaration is sporty but still possible."
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