The scheme, meant to avoid the need for customs controls on the island of Ireland, would see Northern Ireland remain politically in a customs union with the EU but it would be administered by the UK.
But the DUP's deputy leader Nigel Dodds insisted the last-ditch plan "cannot work".
Questioned by the press Friday, Johnson refused to say that Northern Ireland would leave the EU Customs Union, as he previously insisted, replying vaguely, "The whole of the United Kingdom will be able to take full advantage of Brexit". "There is a lot of stuff coming from Brussels, pushed by the Europeans in the last hours, but one thing is sure: Northern Ireland must remain fully part of the United Kingdom customs union".
Barnier's team has always been in "technical talks" with British officials, but without making enough progress towards a treaty text that could forestall a chaotic "no-deal" Brexit on Oct 31.
Now Irish media are suggesting that the "pathway to a possible deal" most likely involves concessions from Mr Johnson with regards to the border, and that the UK/EU border will instead be in the Irish Sea, between Ireland and the UK, instead of on the island between Northern Ireland and the Republic.
While the DUP agree that they do not want checks at the border with Ireland, they say they will not accept any change in ties between Northern Ireland and the rest of the UK.
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Seeking to ensure that Johnson's minority government is not able to hold a general election on his terms, with the Tories claiming to represent the "will of the people" on Brexit, as defined by the 2016 referendum result, they have stepped up their plotting.
The Prime Minister has so far refused to say if he has backed down over the Irish border issue.
On Friday, leading European Union figures said that there was a basis for more talks following new proposals made by Prime Minister Boris Johnson at a meeting Thursday with Irish Taoiseach Leo Varadkar.
With the British Parliament and government both rejecting the withdrawal treaty negotiated by Theresa May, and the European Union rejecting a rehash of the treaty with some alternative arrangements for the customs border between EU member-state the Republic of Ireland and the British province of Northern Ireland put forward by Boris Johnson, the United Kingdom is now headed for a clean break with the bloc - or, if anti-Brexit MPs have their way, yet another Brexit delay.
Teams from both sides started work yesterday to explore whether they can arrive at the basis of an accord ahead of a summit of European Union leaders that begins on Thursday.
MPs pushing for a fresh referendum are expected to try to secure one next weekend when parliament will sit on a Saturday for the first time in nearly 40 years.