Prime Minister Boris Johnson and his Irish counterpart Leo Varadkar are holding further talks to discuss the UK's Brexit proposals.

Downing Street said the meeting would 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.

Labour leader Jeremy Corbyn said the "behaviour and language" used by the PM "suggests he's not going to reach an agreement with the European Union".

But Northern Ireland Secretary Julian Smith called for MPs and commentators to "step back" and "give this meeting space".

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The meeting between the two leaders is taking 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 suggested 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.

Former Tory chancellor Philip Hammond accused Mr Johnson of tabling a proposed deal that "is not going to fly, and was never going to fly".

"Much of it is stuff which has already been tried with the EU in the past and rejected in the past," he told BBC Radio 4's Today programme.

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"

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."

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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 had "yet to be developed".