The UK government has been warned to stick to its commitment to avoid a hard border between Northern Ireland and the Republic.
Irish Foreign Minister Simon Coveney said a time-limited arrangement – or one that could be unilaterally ended by the UK – would never get EU backing.
The border issue is the main barrier to progress between the two sides.
With time running out, Theresa May, who briefs her cabinet on Tuesday, has to get both the EU and her MPs on side.
The UK is due to leave the EU in March, and although 95% of the deal is said to be complete, the tricky bit is proving to be how to honour the commitment by both sides to guarantee no new hard border in Ireland.
It is an issue because after Brexit it will become the UK’s land border with the rest of the EU, which has a single market and customs union so products do not need to be checked when they pass between member states.
There have been warnings that a hard border would undermine the peace process in Northern Ireland.
But unless negotiators can make decisive progress on how to guarantee no new visible checks, a special summit to finalise the UK’s withdrawal will not take place.
Tory Brexiteers are concerned the UK could end up locked in a customs union with the EU without a fixed end point.
Writing in The Sun, former foreign secretary Boris Johnson said this would be an “absolute stinker” of a deal and warned of a “surrender to Brussels” with the UK staying tied to EU rules in years to come.
But Mrs May has insisted that any arrangement would be “strictly time limited”.
This, however, is not the view of the EU.
On Twitter, Mr Coveney said a “time-limited backstop” would not “deliver on previous UK commitments”.
“Still necessary to repeat this, it seems,” added the EU’s deputy chief negotiator, Sabine Weyand.
But the EU’s suggestion of an arrangement that is specific to Northern Ireland has been ruled out by Mrs May who says it would undermine the integrity of the UK by creating a new border down the Irish Sea.
BBC political editor Laura Kuenssberg said Mr Coveney’s comments mattered because there is “deep unease” among Tories about signing up to something without a “best before date”.
The two sides are not going to “suddenly find a Holy Grail”, she said – so “this week has to be decisive”.
Lawyers call for referendum
Meanwhile, 1,400 lawyers have signed a letter calling for another EU referendum to be held.
Among the signatories of the letter are Labour peer Baroness Kennedy QC, former Court of Appeal judge Konrad Schiemann and David Edward, a former judge at the European Court of Justice.
They say questions over the validity of the 2016 vote mean it should not be the public’s final word, any more than the 1975 referendum on membership of what was then the European Economic Community.
In the earlier referendum, voters faced a clear choice between alternatives once negotiations had been completed, the lawyers said.
By contrast, during the 2016 vote, “the nature of the negotiation process and its outcome were unknown”, said the letter.
“Voters faced a choice between a known reality and an unknown alternative. In the campaign, un-testable claims took the place of facts and reality.”
The UK government has said asking the public to vote again would be a betrayal of the public’s trust after the result of the referendum in 2016.
A spokeswoman for the Department for Exiting the European Union said that the government was confident of a “mutually advantageous” deal with the EU.
“The people of the United Kingdom have already had their say in one of the biggest democratic exercises this country has ever seen and the Prime Minister has made it clear that there is not going to be a second referendum,” she said.
Ireland warns UK over post-Brexit border issue