Theresa May has faced warnings that a Brexit deal which involves a hard border in Ireland would be unacceptable from both the Irish Republic and her allies in the Northern Irish DUP.
Ireland's European Commissioner Phil Hogan urged the Prime Minister to change course from a hard Brexit that would see the country leave the single market and customs union.
The country has threatened to use its veto to hold the start of trade talks with the EU if Mrs May cannot find a solution.
It came hours after the DUP leader Arlene Foster said the party would not tolerate any attempt to put barriers between Northern Ireland and the rest of the UK after Brexit.
Mr Hogan slammed the UK delegations "blind faith" in future trade talks and warned that his country will "continue to play tough to the end" over the issue.
"I continue to be amazed at the blind faith that some in London place in theoretical future free trade agreements," he told the Observer.
"The best possible FTA [Free Trade Agreement] with the EU will fall far short of being in the single market. This fact is simply not understood in the UK."
Mrs Foster, whose party is holding up Mrs May's Government on key votes, said that ensuring a soft border within Ireland was among her party's red lines.
She told her party conference on Saturday: "We will not support any arrangements that create barriers to trade between Northern Ireland and the rest of the United Kingdom or any suggestion that Northern Ireland, unlike the rest of the UK, will have to mirror European regulations."
The creation of a soft border within Ireland was a key step in moving on from the violence of the Troubles, but Brexit raises the prospect of checkpoints being raised once again.
The most feasible way to avoid this is by remaining inside the single market or customs union - but Mrs May has said that she wants a clean break from the bloc.
She is facing pressure from hard Brexiteers within her party and her own cabinet on one side, while Ireland pushes her in the other direction.
International Trade Secretary Liam Fox said the final decisions on the border could not be made until a UK-EU trade deal had been agreed.
"We can't come to a final answer to the Irish question until we get an idea of the end state," he told Sky News's Sunday with Niall Paterson.
That was criticised by Labour Shadow Chancellor John McDonnell, who said the comments were worrying.
"One thing that we don't want to do is jeopardise any movement quickly, because we need movement to enable us to get into the proper trade negotiations," he told Peston on Sunday.
Meanwhile, the Scottish Conservative leader Ruth Davidson today insisted that it was not the case that a hard Brexit would have to lead to a physical border.
"I think it's a false dichotomy. I think the Prime Minister has made it absolutely clear in her Florence speech that she doesn't want to see a return to borders of the past," she told the BBC's Andrew Marr show.
"This is all subject negotiation."
Ms Davidson added that the Government accepts that the issue of an Irish border was a "really difficult" aspect of Brexit talks but insisted the UK should not be given an "off-the shelf solution".
"We are a different case to any other country that interacts with the EU," she said.
Meanwhile it emerged the UK could be required to follow new rules implemented by the EU during a Brexit transition period.
The position set out by Michel Barnier in leaked documents would make the application of new EU rules a condition of a transitional deal, meaning Britain could be subject to further Brussels' regulations for about two years after leaving the bloc.
The PrimeMinister was also under pressure not to water down a red line on the EuropeanCourt of Justice's jurisdiction ending when the UK leaves the EU.
The Sunday Telegraph reported that Eurosceptic ministers and backbenchers have expressed alarm at a plan being pushed in the Cabinet's Brexit sub-committee for the UK to agree to a system of "voluntary referral" of cases to the Luxembourg court.