Brexit Secretary Stephen Barclay is to hold talks with the EU’s chief negotiator, Michel Barnier, later.
It comes after European Commission President Jean-Claude Juncker said a new Brexit deal could still be reached before the 31 October deadline.
Prime Minister Boris Johnson said he did not want to “exaggerate progress” but some was being made.
However, Ireland’s foreign minister, Simon Coveney, said there was still a “big gap” between the two sides.
Ahead of the meeting in Brussels, Mr Barclay warned the EU against a “rigid” approach and suggested the final details of an alternative to the Irish backstop may not need to be resolved until the end of 2020.
The backstop is the controversial policy aimed at preventing the return of a hard border on the island of Ireland.
It was a key sticking point in former Prime Minister Theresa May’s attempts to get Parliament to back her withdrawal agreement, which was rejected three times by Parliament.
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Mr Johnson, who has said he wants to leave the EU – preferably with a deal – on 31 October, has urged the EU to scrap the backstop.
The EU has asked for alternative proposals and had previously criticised the UK for not putting any plans in writing.
Mr Juncker told Sky News that he had no “emotional relationship” with the backstop, adding: “If the objectives are met – all of them – then we don’t need the backstop.”
He said his meeting with Mr Johnson in Luxembourg on Monday was “rather positive”, and “we can have a deal” in the next few weeks.
The cautiously optimistic tone was shared by Mr Johnson, who said: “I don’t want to exaggerate the progress that we are making, but we are making progress.”
The prime minister said the UK needed to leave in a way that allowed 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 was protected.
“We think we can do that,” Mr Johnson said. “We think we can solve that problem and I think we are making some progress.”
He added: “Let’s see where we get. It is vital whatever happens that we prepare for no-deal, and we will be ready for no-deal on 31 October. We have got to do both things at once.”
BBC political correspondent Nick Eardley said No 10 had been “heartened” by the EU’s agreement to accelerate talks and the willingness to explore ideas Britain has to replace the Irish backstop.
But he added that a breakthrough in the immediate term was unlikely and a positive tone in the Brexit process did not always lead to positive outcomes.
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.
It came 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.
Irish prime minister Leo Varadkar, who will be among those at the summit, said he would try to get a deal with Mr Johnson.
However, his deputy, Mr Coveney, said: “There’s still a big gap between what the British government has been suggesting that they’re looking for and what Ireland and the EU need in terms of getting a deal, and in order to close that gap we need to get credible proposals from the British government, which we simply haven’t received yet.”
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Thursday also saw the final day of the legal battle over Mr Johnson’s decision to suspend Parliament at the UK’s Supreme Court.
The prime minister prorogued Parliament earlier this month for five weeks, with MPs not scheduled to return until 14 October.
The President of the Supreme Court, Lady Hale, said the justices would announce a decision early next week.
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