Boris Johnson has said he is “cautiously optimistic” about the prospects of a new Brexit deal but there is growing impatience among EU leaders who are demanding “concrete” proposals from the British side.

– What discussions have been taking place?

The Prime Minister held his first meeting with European Commission president Jean-Claude Juncker in Luxembourg on Monday when they agreed to step up official level contacts.

Mr Johnson has also been meeting key EU leaders, including Ireland’s Leo Varadkar, and is expected to have further meetings – including with European Council president Donald Tusk and Germany’s Angela Merkel – next week in the sidelines of the UN General Assembly in New York.

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Boris Johnson has held talks with European Commission president Jean-Claude Juncker (Stefan Rousseau/PA)

– What progress have they been making?

Mr Johnson has said the basic shape of a deal to resolve the vexed issue of the Northern Ireland backstop is becoming clear.

However, on the EU side, there is frustration that the UK side has yet to put forward any formal proposals in writing – something the British appear reluctant to do until they are sure they will receive a favourable reception.

– Why is the backstop such an issue?

The backstop is designed as a temporary measure to ensure there is no return to a hard border between Northern Ireland and the Republic while talks on a new trade deal are taking place after Britain has left the EU.

Under its provisions, the whole of the UK would effectively remain part of the EU customs union while Northern Ireland would also be required to follow some rules of the single market.

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Maintaining an open border between Northern Ireland and the Republic is key to any Brexit deal (Brian Lawless/PA)

The EU argues that it is necessary to keep the border open – in line with the provisions of the Good Friday agreement – while preserving the integrity of the single market which the UK is leaving.

However, Mr Johnson says it is “anti-democratic”, potentially tying Britain indefinitely to EU rules with no say on any changes or new regulations.

– So what is the alternative?

Details are sketchy, but they appear to involve re-working the backstop – although the name would almost certainly have to be changed – with Northern Ireland becoming a special economic zone.

It would mean Northern Ireland continuing to follow EU rules on agriculture, foodstuffs and some other products while maintaining the common electricity market with the Republic as well as the common travel area.

It would, however, remain outside the customs union, with checks on other goods being “de-dramatised”, taking place away from the border.

– Will it work?

The EU side remains sceptical, insisting it has yet to see any workable, legally binding proposals that would enable customs checks to take place away from the border.

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Boris Johnson meeting Taoiseach Leo Varadkar in Dublin (Niall Carson/PA)

It also implies new checks on goods travelling between Northern Ireland and the rest of the UK – effectively imposing a border in the Irish Sea – something the DUP and as well as many Tories are determined to resist.

There is also a deep reluctance in Brussels to sign up to any new agreement unless they can be confident it will get through Parliament after Theresa May’s deal was rejected three times by MPs.

– So what are the chances of Parliament signing up to such a plan?

On the plus side for Mr Johnson, the DUP – while insisting it has not dropped any of its “red lines” – has been sounding more emollient of late.

However, he is still likely to face a rebellion from some hardcore Tory Brexiteers – in which case he is likely to need the support of some Labour MPs who believe Britain should leave the EU but fear a no-deal Brexit.

While there have been some positive noises from MPs in heavily Leave-voting areas, it remains to be seen whether they would vote in sufficient numbers to get a Tory prime minister off the hook.

– So what are the chances of getting a deal in place before Britain’s scheduled departure on October 31?

While it is still possible, time is running out and many seasoned observers in both Westminster and Brussels believe the odds are stacked against Mr Johnson.

In the end, it may come down to whether the political will exists on both sides of Channel to bridge the gap between them.