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 added that Prime Minister Boris Johnson and European Commission President Jean-Claude Juncker were “clear” that was what they wanted.
The commission said it was “open” to proposals from the UK.
The deadline for the UK to exit the EU is 31 October.
On Thursday, Mr Juncker said a new Brexit deal could still be reached by then.
Mr 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.
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.
Mr Johnson, who has said he wants to leave the EU – preferably with a deal – by the current deadline of 31 October, has urged the EU to scrap the backstop.
But the EU has asked for detailed alternative proposals.
Following the latest meeting, Mr Barclay said: “We both want to see a deal.”
He added that “a clear message has been given both by President Juncker and the prime minister”.
Mr Barclay also said both sides were “working hard on that”.
The European Commission issued a statement, saying Mr Barnier and Mr Barclay had discussed the “state of play” on Brexit.
It added: “It is essential that there is a fully workable and legally operational solution included in the withdrawal agreement. We remain willing and open to examine any such proposals that meet all the objectives of the backstop.”
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.
Mr Johnson will hold more talks with European leaders at a UN summit in New York next week.
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.