Brexit Secretary Stephen Barclay has said the UK and EU share a "common purpose" in reaching a new withdrawal deal, after a meeting in Brussels with chief EU negotiator Michel Barnier.
He said they had had "serious detailed discussions" and things were "moving forward with momentum".
Mr Barnier said it had been a "cordial" meeting, but "lots of work has to be done in the next few days".
The deadline for the UK to exit the EU is 31 October.
On Thursday, European Commission President Jean-Claude Juncker said a new Brexit deal could still be reached by then.
Prime Minister Boris Johnson said some "progress" was being made, although it was important not to "exaggerate" this.
But Irish Foreign Minister Simon Coveney told the BBC on Friday that there was a "wide gap" between the UK and the EU, with Brussels "still waiting for serious proposals" from London.
And a leaked memo from the European Commission said the UK had confirmed its proposals for replacing the Irish backstop "do not amount to legally operational solutions and would have to be developed during the transitional period".
The memo also said the UK's proposals did not avoid a hard border on the island of Ireland, or preserve the integrity of the single market.
However, a UK government source insisted their proposals were "serious and workable" and would avoid a hard border.
"As for the Commission, two months ago they said we couldn't reopen the withdrawal agreement and there was absolutely no alternative to the backstop - now we are having detailed discussions.
"Leaks from Brussels on Twitter are par for the course, you can set your watch by them.
"What we're focused on is actually getting a deal in the room - we trust they'll do the same."
The backstop - the policy aimed at preventing the return of a hard border on the island of Ireland after Brexit - has proved the biggest point of contention in EU-UK talks so far.
It was a major sticking point in former Prime Minister Theresa May's attempts to get Parliament to back her withdrawal agreement, which was rejected three times by MPs.
Enough oomph for a breakthrough?
Amid the white noise of Brexit, there has only ever been one question that really matters: is there any sign that things might be about to change?
Meetings get excitedly talked up in advance (I plead guilty). Anodyne statements follow shortly afterwards.
In short, the EU says the UK has not yet provided enough detail on a plan to replace the so-called backstop, to keep the border on the island of Ireland open under all circumstances.
The UK says it is dreaming up all sorts of ideas.
So what happens next? New York. A shindig at the United Nations next week - and with it the chance for the prime minister to have one-on-one chats with the likes of President Macron of France and Chancellor Merkel of Germany.
Could they provide the oomph needed to knock together a deal?
It still looks less likely than likely, but who knows?
Mr Johnson, who has said he wants to leave the EU - preferably with a deal - by 31 October, has urged the EU to scrap the backstop.
But the EU has asked for detailed alternative proposals.
Following the latest meeting, which overran, Mr Barclay said: "There's a common purpose in Dublin, in London and here in Brussels to see a deal over the line."
He added that the two sides had been "getting into the detail" and that more "technical" discussions would happen next week.
Mr Johnson and European Council President Donald Tusk are also expected to hold talks when the United Nations General Assembly takes place in New York next week.
Mr Barclay said this "underscores the purpose there is on both sides to get a deal and that is what we are working very hard to secure".
He added that "a clear message has been given both by President Juncker and the prime minister" and both sides were "working hard".
Also speaking after the meeting, Mr Barnier said he was not optimistic or pessimistic but "still determined".
"Brexit is a school of patience but we are still ready to reach an agreement," he said.
He said that any proposal from the UK to replace the backstop "must reach all the objectives of the backstop".
These were to "protect the peace in Ireland, to protect the all=Ireland economy, and also to protect the consumers and the businesses of the EU and the single market", Mr Barnier said.
The prime minister has said the UK needs to leave in a way that allows it to "do things differently" and "not remain under the control of the EU in terms of laws and trade policy".
But he also reiterated the need to ensure no hard border returned to Northern Ireland, and the Good Friday Agreement - which helped bring an end to the Troubles - was protected.
On Thursday, the UK government said confidential documents that "reflect the ideas the UK has put forward" on Brexit had been shared with the EU.
This happened after Finland's prime minister said Mr Johnson had 12 days to set out his Brexit plans to the EU - although a government source said the development was not in response to those remarks.
Thursday saw the final day of the legal battle over Mr Johnson's decision to prorogue - suspend - Parliament at the UK's Supreme Court.
Its president, Lady Hale, promised a decision early next week.