EU countries have agreed to "intensify" Brexit talks with the UK over the next few days.
The development comes after a meeting in Brussels between Brexit Secretary Steve Barclay and EU chief negotiator Michel Barnier, described by both sides as "constructive".
But UK Prime Minister Boris Johnson said there was "a way to go" before a deal could be reached.
The UK is due to leave the EU at 23:00 GMT on 31 October.
A European leaders' summit next week is seen as the last chance to agree a deal before that deadline.
Mr Johnson put forward revised proposals for a deal last week, designed to avoid a hard border on the island of Ireland after Brexit.
Speaking on Friday, he said there was not "a done deal", adding: "The best thing we can do now is let our negotiators get on with it."
In a statement, the European Commission said: "The EU and the UK have agreed to intensify discussions over the coming days."
BBC Brussels correspondent Adam Fleming said that, although there would be a "measure of confidentiality", EU states would still be briefed on any developments.
In Paris, French President Emmanuel Macron was asked if there was hope of a deal, replying: "Let's wait for the next few hours."
Leader of Northern Ireland's Democratic Unionist Party Arlene Foster said "anything that traps Northern Ireland in the EU, whether single market or customs union, as the rest of the UK leaves will not have our support - the prime minister is very mindful of that".
Support from DUP MPs will be key to getting a deal through Parliament.
Meanwhile, the pound rose to a three-month high against other major currencies, amid increased investor optimism over an agreement.
Following his "technical" discussion with Mr Barclay in Brussels, Mr Barnier told reporters: "Be patient."
He added: "Brexit is like climbing a mountain. We need vigilance, determination and patience."
Earlier this week, several senior EU figures played down the chances of Brexit happening with a deal agreed to by both sides.
But on Thursday, Mr Johnson and Mr Varadkar held a one-to-one discussion - described by both as "very positive and very promising".
Mr Varadkar said talks were at a "very sensitive stage" but he was now "convinced" the UK wanted a deal, adding: "I do see a pathway towards an agreement in the coming weeks."
What are the PM's border plans?
Under Mr Johnson's proposals, which he calls a "broad landing zone" for a new deal with the EU:
- Northern Ireland would leave the EU's customs union alongside the rest of the UK, at the start of 2021
- But Northern Ireland would continue to apply EU legislation relating to agricultural and other products, if the Northern Ireland Assembly approves
- This arrangement could, in theory, continue indefinitely, but the consent of Northern Ireland's politicians would have to be sought every four years
- Customs checks on goods traded between the UK and EU would be "decentralised", with paperwork submitted electronically and only a "very small number" of physical checks
- These checks should take place away from the border itself, at business premises or at "other points in the supply chain"
With less than three weeks to go until the Brexit deadline, Mr Johnson continues to insist the UK will leave the EU with or without a deal at the end of the month.
That is despite the so-called Benn Act - passed by MPs last month - demanding he request a delay to the Article 50 deadline from the EU until January 2020 if a deal, or a no-deal exit, has not been approved by Parliament before 19 October.
Should any new deal be reached between the EU and UK, it will still have to be approved by MPs, as well as the European Parliament.
MPs will be called to Parliament for a special Saturday sitting following next week's EU summit, the first since 1982, when the House of Commons was summoned after the invasion of the Falkland Islands.
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.