European Council president Donald Tusk has warned Theresa May she must satisfy Irish demands that there will be no "hard border" between the Republic and the North if the Brexit negotiations are to move forward.
Following talks with Taoiseach Leo Varadkar in Dublin, Mr Tusk said it was now up to the UK Government to come forward with a "credible commitment" to deal with the border issue.
He said he had agreed to consult Mr Varadkar on any British proposal before deciding whether to recommend EU leaders give the green light to the second phase of Brexit negotiations, including talks on a free trade deal, to begin at their summit on December 14 and 15.
"If the UK offer is unacceptable for Ireland, it will also be unacceptable for the EU," he told a joint news conference with Mr Varadkar.
"I realise that for some British politicians this may be hard to understand, but such is the logic behind the fact that Ireland is an EU member while the UK is leaving.
"That is why the key to the UK's future lies in some ways in Dublin, at least as long as the Brexit negotiations continue."
Mr Tusk has given Mrs May until Monday to come forward with an improved offer on the terms of the UK's withdrawal, including the "divorce bill" and future citizens' rights, as well as the Irish border, if there is to be progress at the December summit.
The Prime Minister is due to travel to Brussels on Monday when she is expected to explain the latest British position at a lunch with European Commission president Jean-Claude Juncker.
But while the two sides appear to be moving closer on the issues of citizens' rights and the divorce settlement, with Mrs May reportedly ready to pay around £40 billion to settle the UK's outstanding obligations, the border issue has emerged as a major stumbling block.
The Irish government has warned that as long as the UK remains committed to leaving the single market and the customs union, the return of some form of physical border controls is inevitable.
Mr Varadkar said that while there had been "good progress" on the Ireland issue, the UK now needed to present "credible, concrete and workable solutions" which guarantee there will be no hard border.
"I believe that with the right engagement and the right political will, we can reach an agreement on the way ahead," he said.
"I am also prepared to stand firm with our partners if the UK offer falls short on any of those three issues, including the Irish ones."
Earlier Irish foreign minister Simon Coveney said Dublin was looking for an "agreed wording" with the UK setting out the parameters within which a solution to the border issue could be found.
In particular, he said they were seeking ways to avoid "regulatory divergence" between the North and the Republic after Brexit.
"If you have different standards in terms of food safety, animal welfare, animal health, if you have different standards in relation to medical devices and the approval of drugs, how then can you maintain practical north/south co-operation as we have it today," he told the BBC Radio 4 Today programme.