Negotiators from the UK and EU are having what has been described as "intense technical discussions" in an attempt to agree a new Brexit deal.
About a dozen British officials, including the UK's EU adviser David Frost, are taking part in the talks at the EU Commission in Brussels.
The meetings are expected to continue through the weekend.
But European Council President Donald Tusk has suggested there is only the slightest chance of an agreement.
The UK is due to leave the EU at 23:00 GMT on 31 October and a European leaders' summit next Thursday and Friday is seen as the last chance to agree a deal before that deadline.
Prime Minister Boris Johnson's revised proposals - designed to avoid concerns about hard border on the island of Ireland after Brexit - were criticised by EU leaders at the start of last week.
However, on Thursday, Mr Johnson and the Irish prime minister Leo Varadkar held talks and said they could "see a pathway to a possible deal".
BBC Europe reporter Gavin Lee said there is no scheduled timetable for the discussions in Brussels and neither the UK or EU are offering any detail yet on the apparent common ground that has been found on the Irish border.
Our correspondent said the first public announcement on the talks may come on Monday, after the EU's 27 ambassadors have been updated on the progress so far.
Meanwhile, shadow Brexit secretary Sir Keir Starmer says Labour would take action through the courts if Mr Johnson tries to push through a no-deal Brexit.
Addressing the Co-operative Party conference in Glasgow, Sir Keir said if the PM did not secure a deal at the EU summit on 17 and 18 October, he must comply with the so-called Benn Act passed by MPs in September, which requires him to seek a further delay.
"If he doesn't, we'll enforce the law - in the courts and in Parliament. Whatever it takes, we will prevent a no-deal Brexit," he said.
'Mood music positive'
This weekend's talks in Brussels follow a meeting on Friday between Brexit Secretary Steve Barclay and EU chief negotiator Michel Barnier, described by both sides as "constructive".
In a statement issued later, the European Commission said: "The EU and the UK have agreed to intensify discussions over the coming days."
Speaking on BBC Radio 4's Today programme, Culture Secretary Nicky Morgan reiterated that "lots of details" needed to be worked out between both parties but said the "mood music" on negotiations "seems positive".
She added that "speculation doesn't really help" and politicians needed to "stand back and give those negotiations and discussions the best chance of succeeding".
On Friday, Mr Tusk said he had received "promising signals" from the Irish PM, before adding: "Of course there is no guarantee of success and time is practically up, but even the slightest chance must be used".
Mr Johnson also acknowledged there was not "a done deal", saying: "The best thing we can do now is let our negotiators get on with it."
Support from Democratic Unionist Party MPs could be crucial to get a deal through Parliament.
But DUP leader Arlene Foster said: "Anything that traps Northern Ireland in the EU... will not have our support."
Brexiteer Sir John Redwood believes Mr Johnson should "table a free trade agreement" which would "unlock" most of the issues around borders and immigration.
He added: "I think the border issue is greatly exaggerated, because it is in the interest of the European Union and Ireland to exaggerate it."
Ms Morgan was asked on the Today programme about reports of Downing Street briefings that the Tories could contest a general election on a no-deal Brexit ticket, if an agreement cannot be reached.
The Loughborough MP - who voted Remain - did not say whether she would contest an election on such a ticket, but said reports that Mr Johnson is preparing to fight a general election on a no deal platform are "wide of the mark".
Timeline: What's happening ahead of Brexit deadline?
Monday 14 October - The Commons is due to return, and the government will use the Queen's Speech to set out its legislative agenda. The speech will then be debated by MPs throughout the week.
Thursday 17 October - Crucial two-day summit of EU leaders begins in Brussels. This is the last such meeting currently scheduled before the Brexit deadline.
Saturday 19 October - Special sitting of Parliament and the date by which the PM must ask the EU for another delay to Brexit under the Benn Act, if no Brexit deal has been approved by Parliament and they have not agreed to the UK leaving with no-deal.
Thursday 31 October - Date by which the UK is due to leave the EU, with or without a withdrawal agreement.