Theresa May has been warned that the EU will block progress in the Brexit talks if the Irish government decides that her proposals for the border with Northern Ireland are “unacceptable”.

The European Council President, Donald Tusk, effectively handed Dublin a veto over progressing the talks onto future trade next month – the Prime Minister’s priority in the negotiations.

“Let me say very clearly. If the UK offer is unacceptable for Ireland, it will also be unacceptable for the EU,” Mr Tusk said at a press conference alongside the Irish Prime Minister Leo Varadkar in Dublin.

He added: “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.”

Mr Varadkar said that while the “best option” to avoid a hard border in Ireland after Brexit – a major sticking point in the negotiations so far – would be for the UK to remain in the customs union, he recognised this is not Ms May’s desire.

Donald Tusk holds talks with Ireland’s Prime Minister Leo Varadkar (AFP/Getty) (AFP/Getty Images)

He added: “As the British Government has ruled that option out it must offer credible, concrete and workable solutions that guarantee that there will be no hard border whatever the outcome of the negotiations.

“As we discussed today, the period between now and the European Council meeting in two weeks’ time will be crucial – indeed, the next couple of days. So we don’t have long, but I believe that with the right engagement and with the right political will we can reach an agreement on the way ahead.”

But Mr Varadkar warned: “I am also prepared to stand firm if the offer falls short.”

The intervention by Mr Tusk and the Irish PM sets the scene for the coming days, in which Ms May will have to offer concrete proposals on the issue of the Irish border within days.

And on Monday, she is expected to meet Jean-Claude Juncker ahead of this month’s crucial EU summit, where it will be decided whether the negotiations can advance to phase two – discussions on a future relationship and trading agreement.

In order to make that decision, EU leaders will have to be satisfied that negotiators have made “sufficient progress” on three key areas, including citizens’ rights after Brexit, the Irish border and the financial settlement – often referred to as the divorce bill.

Donald Tusk with Leo Varadkar (PA)

While a significant breakthrough was reached earlier this week on the divorce bill, the issue of the Irish border has caused a considerable headache for the Prime Minister, who is relying on the Democratic Unionist Party in Northern Ireland to prop up her fragile Government.

On Thursday the DUP threatened to rethink its deal with the Conservatives as Sammy Wilson, one of the party’s 10 MPs in Westminster, said they would not accept any moves that would see Northern Ireland “treated differently than the rest of the UK”.

The stark warning followed reports that further powers could be devolved to Belfast to allow rules in areas such as agriculture and energy to be aligned with the EU, rather the London.

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The move is an attempt to meet the Irish government’s demand for a cast-iron guarantee there will be no return to a hard border with Norther Ireland, which it fears would be inevitable because of new trade barriers.

“If there is any hint that, in order to placate Dublin and the EU, they’re prepared to have Northern Ireland treated differently than the rest of the United Kingdom, then they can’t rely on our vote,” Mr Wilson said.

“They have to recognise that, if this is about treating Northern Ireland differently, or leaving us half in the EU, dragging along behind regulations which change in Dublin, it’s not on.”