The EU's negotiating position is that "sufficient progress" must be made in three key areas before talks can progress to their second phase.
The paper contains the most explicit expression yet by the EU that the only way to avoid a hard border is for the United Kingdom to remain in the customs union and the single market, or at least for Northern Ireland to remain within those structures.
Mr Coveney said the Government's position is "consistent, firm and stubborn" that there should be no "regulatory divergence" post Brexit, in order to allow for a "functioning all-island economy".
"We believe that can be best achieved by continuing to apply the same sort of rules and regulations".
It comes as Mr Barnier met Brexit Secretary David Davis in Brussels today for round six of the negotiations.
Speaking in Jersey, he said there was a precedent for a tailored solution in the Isle of Man, which is a member of the customs union and has free movement of goods and products without being in the EU.
That scenario would effectively require Northern Ireland to remain in the single market and the customs union.
Neither want Brexit to lead to the emergence of a hard border with the Republic of Ireland.
But its articulation of the customs union and single market issues are likely to prompt fierce resistance from the DUP.
They say that a hard border can only be avoided by "meaningful" solutions put forward by the UK.
The Foreign Affairs Minister Simon Coveney has said Britain does not have the right to democratically decide or shape Ireland's future.
"While we welcome the language we get from the British government in the context of north-south challenges. there has always been a scepticism on how we are going to get there in the context of the British approach to Brexit as a whole", he said.
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