SALZBURG - EU President Donald Tusk warned Thursday after talks with Prime Minister Theresa May that Britain's plan for post-Brexit trade arrangements simply "will not work".

After two days of talks in the Austrian city of Salzburg, Tusk told reporters that European Union leaders believe May's so-called Chequers plan for Brexit would undermine the bloc's single market. The Polish former leader said the atmosphere in talks between May and the other 27 leaders had been better than before but that stark differences remained on trade and on the question of the Irish border.

"It must be clear that there are some issues where we are not ready to compromise, first off the four fundamental freedoms, the single market, this is why we remain sceptical of Chequers," Tusk said. "The Irish question remains our priority too and for this we need only goodwill -- which we feel, the atmosphere was better than two or three weeks ago -- but the Irish question needs something more than good intentions," Tusk said.

"We need tough, clear and precise guarantees. This is why we need more time, but we hope to be ready in October," he said, dubbing the planned October 18 EU summit the "moment of truth."

Both Tusk and summit host Austrian Chancellor Sebastian Kurz said a follow-on November emergency summit would only go ahead if enough progress is made by October to justify it.

The meeting in Salzburg, Austria, is the first of three summits in successive months, which Brussels hopes will end with agreement on the terms of Britain's withdrawal from the bloc set for March 29 next year. The Alpine summit began with a warning from EU Council President Donald Tusk that Britain's offer on post-Brexit trade ties and Ireland -- the two sticking points in the talks -- must be "reworked".

May, who is under intense pressure from Brexiteers back home, retorted that she had already made compromises and it was now Brussels' turn.

She told EU leaders that their proposal to "carve away" Northern Ireland from the rest of the United Kingdom, to avoid checks on the border after Brexit, was "not credible".

"The approaches remain very different," Austrian Chancellor Sebastian Kurz told reporters as he arrived for Thursday's talks. "Away from the discussion in the media, behind closed doors, I have the impression that both sides are aware that a solution can only be found if both sides make a move towards one another. "It will nevertheless still be a difficult process." Luxembourg Prime Minister Xavier Bettel added: "It is an agreement, so we both have to find a compromise."

French President Emmanuel Macron insisted Britain must still explain how it intends to avoid physical border checks in Ireland after leaving the EU's single market and customs union.

Both sides have pledged to avoid a "hard" frontier in Ireland, amid fears it could disrupt trade and upset the fragile peace on the island.

Britain has put forward a proposal as part of its Chequers plan for post-Brexit trade ties, but accepts that some fallback plan is needed until this can be agreed. The EU version of the so-called "backstop" would see Northern Ireland alone continue to follow many EU trade rules and regulations -- but London says this would undermine the integrity of the UK.

May held one-on-one talks on Thursday with Irish Prime Minister Leo Varadkar, who had earlier told the BBC: "I don't think we're any closer to a withdrawal agreement than we were in March."

But there were signs of some movement, after the EU's chief Brexit negotiator, Michel Barnier, this week suggested any checks between Northern Ireland and the rest of the UK could be away from the border.

May welcomed his willingness to "find a new solution", and conceded that some checks were already carried out in the Irish Sea, on agricultural products.

A senior British government official also said it would bring forward a backstop proposal for regulatory checks on goods crossing into and from Northern Ireland, raising the possibility of a breakthrough.

However, May repeated her warning that Britain would never agree to any "legal separation of the United Kingdom into two customs territories".

Both sides had been aiming for an October EU summit as the deadline to reach an agreement, to allow time for the deal to be ratified by British and European parliaments before Brexit in March.

But with the talks deadlocked, Kurz confirmed Thursday there would be a final summit in November to clinch the deal.

Over dinner on Wednesday, May had her first opportunity to present to fellow leaders her plan for the post-Brexit trading relationship, which was published in July.

Her proposal to follow EU rules on trade in goods has provoked a fierce backlash among eurosceptics in her Conservative party, renewing speculation of a challenge to her leadership.

The political turmoil has sparked growing calls in Britain for a re-run of the 2016 Brexit vote -- calls echoed by both the Maltese and Czech leaders in Salzburg.

However, May warned fellow leaders it would not happen, nor would Britain consider delaying Brexit to allow more time for negotiations.

"The UK will leave on March 29 next year," she said over dinner, adding: "The onus is now on all of us to get this deal done."