EU chief negotiator Michel Barnier has likened Brexit to "climbing a mountain", after holding talks with Brexit Secretary Steve Barclay.
Mr Barnier urged "patience", as the EU and UK look to reach a deal ahead of a crucial summit next week.
Meanwhile, European Council President Donald Tusk said Irish PM Leo Varadkar had given "promising signals" that an agreement is still possible.
The UK is due to leave the EU at 23:00 GMT on 31 October.
The summit on 17 and 18 October is seen as the last chance for the UK and the EU to agree a deal ahead of the Brexit deadline.
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."
BBC Brussels correspondent Adam Fleming said he suspected the best outcome for the UK that could emerge from the Barclay-Barnier meeting would be the EU agreeing to restart fuller and "more intense" negotiations.
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, UK Prime Minister Boris Johnson and Mr Varadkar held a one-to-one discussion - described by both as "very positive and very promising" - at Thornton Manor, Merseyside.
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."
But he refused to be drawn on what "concessions" had been made by either side during his meeting with Mr Johnson, while Downing Street has not commented on Irish press reports suggesting "significant movement" on the UK prime minister's part.
Speaking on a visit to Cyprus on Friday, Mr Tusk said the UK had not "come forward with a workable, realistic proposal", but added he had received "promising signals" from Mr Varadkar that a deal was still possible.
"Of course there is no guarantee of success and time is practically up, but even the slightest chance must be used," he added. "A no-deal Brexit will never be the choice of the EU."
With three weeks to go until the Brexit deadline, the prime minister 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 has not been agreed before 19 October.
Should any new deal be reached between the EU and UK, it will still have to be approved by MPs in Parliament.
The issue of Northern Ireland's consent - and how it is achieved - for post-Brexit arrangements has emerged as a key factor in EU-UK negotiations.
But Northern Ireland Secretary Julian Smith said there was "a range of options" for finding consent for the proposals.
He also denied any "one community" would have a veto over Brexit plans, following concern from some NI parties that the PM's proposals could give the Democratic Unionist Party exactly that.
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"
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.