DUP leader Arlene Foster has repeated her insistence that there should be no customs or regulatory barriers between Great Britain and Northern Ireland, and said she had made this clear to the British Prime Minister Theresa May.
For industrial goods, he said it could be done by "market surveillance authorities" in company premises but he added that checks on animals and animal-derived products would still have to take place on the border.
But he also set out key objections to her proposed new trading relationship which involves a UK-EU customs partnership and a "common rulebook" for goods travelling across the border.
Sources in the party - whose 10 MPs are propping up the PM in power - have warned it would be "unacceptable" for Northern Ireland and mainland Britain to be "treated differently".
"The whole point about checks is that there is a difference" she said. They know where we stand.
In a statement to the Commons updating MPs following a summit of EU leaders in Salzburg last month, Brexit Secretary Dominic Raab denied the United Kingdom would stay in the customs union indefinitely, and said the government would "press our case" for an outcome that will "preserve the integrity of the UK".
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However, the European Union says that means there must be some checks on goods coming into Northern Ireland in order to prevent those that do not meet European Union standards from going across in the Republic of Ireland and then reaching the continent.
There are usually checks on goods coming into the EU's customs territory from outside countries, but the bloc has agreed not to carry out such checks between Northern Ireland and the Republic of Ireland after Brexit in order to safeguard the peace process.
Barnier told small business leaders that the talks were "continuing intensively this week, day and night, with the aim. of having a deal within reach, if we follow through to the end of this negotiations, on October 17".
Conservative MP Steve Baker said on Tuesday that at least 40 of his colleagues are ready to vote against a deal which included that solution because it would represent a "half in, half out Brexit".
Following the College of Commissioners meeting, two commissioners stressed the importance of continued UK/EU co-operation on security after Brexit. There remain big issues to work through and, as the PM has said, this will require movement on the European Union side.
If 40 of her lawmakers voted against a possible deal, the fate of the government and exit process would depend on the opposition Labour Party, which has indicated it will vote against nearly any deal May might secure.