Prime Minister Boris Johnson and his Irish counterpart Leo Varadkar will meet for talks later to discuss the UK’s Brexit proposals.
Downing Street said the meeting was to allow “detailed discussions” on the process of securing a deal.
Mr Johnson has said he remains “cautiously optimistic” about a deal.
But on Wednesday EU leaders accused the UK of putting forward untested ideas, adding that progress towards a new deal had been limited.
A similar sentiment was expressed on Tuesday by Mr Varadkar, who suggested it would be “very difficult” for the UK and the EU to reach a Brexit agreement before the 31 October deadline.
The meeting between the two leaders will take place in the north-west of England, with Mr Johnson still insisting 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.
On Wednesday, Business Secretary Andrea Leadsom said the PM was gearing up to bypass legal obstacles to a no-deal Brexit by sending one letter requesting an extension and, in the same instance, submitting a second memo telling European leaders he does not want one.
Asked on ITV’s Peston programme whether the idea of sending two letters to the EU was a possible loophole, Ms Leadsom replied: “Absolutely.”
Mr Johnson put forward fresh proposals for a Brexit deal last week, but Mr Varadkar has said “big gaps” remain between the UK and the EU.
Updating MEPs on the state of talks, the EU’s chief negotiator, Michel Barnier, said he believed “with goodwill” on both sides there could be an agreement in the run-up to the EU summit later this month.
But he added: “To put things very frankly and to try to be objective, we are not really in a position where we are able to find an agreement.”
As it stood, he continued, the UK was proposing replacing an “operable, practical and legal solution” to avoid a hard Irish border with “one that is simply a temporary solution”.
Mr Barnier said the UK’s suggested alternative to the Irish backstop – which would see customs checks conducted away from the border at business premises or electronically – “had not been tested” and was “largely based” on exemptions for small businesses and technology that “has yet to be developed”.
Timeline: What’s happening ahead of Brexit deadline?
Thursday 10 October – Boris Johnson meets Irish leader Leo Varadkar in the north-west of England for “detailed discussions” over the UK’s Brexit proposals.
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 – 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.