The British Prime Minister Theresa May is defiantly sticking by her strategy for Brexit, despite the EU insisting a key element of her plan "will not work".

The President of the European Council, Donald Tusk, earlier warned that her plan for future trade with the EU is unworkable. 

He was speaking at the end of the informal meeting of EU leaders in Salzburg on Thursday afternoon.

He once again confirmed that there will be no Withdrawal Agreement without a "solid, operational and legally binding" backstop solution preventing any hard border on the island of Ireland.

"We continue to fully support Michel Barnier in his efforts to find such a model," he said.

It comes after Mrs May claimed that the Brexit proposals agreed by the British cabinet at her Chequers country estate were the "only serious credible" way to avoid a hard border.

She also repeated her opposition to an EU proposal for an Irish border backstop - in the event of a no-deal Brexit - as it would "divide the UK into two customs territories".

Instead, Mrs May said she would bring forward new proposals of her own "shortly".

She also raised the prospect of the UK leaving the EU without an agreement, adding: "Let nobody be in any doubt: as I have always said, we are preparing for no deal.

"So that if we get to the position where it's not possible to reach a deal, then the British people can be confident that we will have done what is necessary to ensure we make a success of leaving the EU."

"Unworkable"

Mr Tusk also took aim at other elements of the Chequers plan, noting that its vision for future economic cooperation between the two blocs is unworkable.

"Everybody shared the view that while there are positive elements in the Chequers proposal, the suggested framework for economic cooperation will not work - not least because it risks undermining the Single Market," he said.

He said the "moment of truth" for the talks will come at the October European Council Summit.

He said that if enough progress is made, leaders will use the summit to "decide whether conditions are there to call an extraordinary Summit in November to finalise and formalise the deal."

Germany and France

German Chancellor Angela Merkel and French President Emmanuel Macron also gave a thumbs down to Mrs May's proposals.

Ms Merkel warned there is "still a lot of work to do" before a future trade agreement is reached.

While Mr Macron said: "We all agreed on this today, the proposals in their current state are not acceptable. The Chequers plan cannot be take it or leave it."

Mr Macron also attacked those who "predicted easy solutions" for leaving the EU as "liars", adding: "What's more, they left the next day so they didn't have to handle it."

Hard border

Earlier, the Taoiseach Leo Varadkar warned that any deal must avoid "any new barriers to the movement of goods, any new barriers to trade, any new barriers to the movement of people" in Ireland.

"I think we need to redouble our efforts over the next couple of weeks to make sure that we have a deal and that deal, of course, would include three elements," he said.

"A Withdrawal Agreement providing for the UK to leave with a transition period, a protocol on Ireland and Northern Ireland - making sure that under all circumstances there won't be a hard border on the island of Ireland - and also a joint political declaration outlining what that new relationship is going to look like."

He was speaking after a bi-lateral meeting with Mrs May on the margins of the gathering.

He said he had a 'very good meeting' with the UK leader and her team.

"Time is running short," he said. "There's proposals for another summit in October, and then in November."

"Ireland as a country obviously wants to avoid a no-deal scenario... we want to avoid a no-deal Brexit. But we are preparing for that."

Leo Varadkar smiles when arriving at the informal EU summit in Salzburg, Austria | Image: Kerstin Joensson/AP/Press Association Images

He warned however that it is for the EU and UK to negotiate the details of the deal.

"Prime Minister May and I have conversations about this all the time but this is a negotiation between the European Union and the United Kingdom," he said.

"Ireland is staying at the heart of Europe, we're staying in the single market and the euro, and we helped defend those great projects that have brought enormous prosperity across the continent of Europe.

"This ultimately is going to be a new treaty, a new negotiation between the EU and UK."

Chequers

The UK's decision to leave the bloc is set to be a key focus during the second day of the summit, with EU leaders also looking at issues such as migration and internal security.

Mrs May used a dinner in Austria to say there would be no second referendum and no delay to the withdrawal date in March.

Speaking on Wednesday, Mrs May also insisted that her Chequers proposals are the only way to avoid a hard border in Ireland.

She said: "If we're going to achieve a successful conclusion, then just as the UK has evolved its position the EU will need to evolve its position too.

"I'm confident that with goodwill and determination we can agree a deal that's right for both parties."

She urged EU leaders not to rip the UK apart with "unacceptable" Brexit demands and claimed that keeping the North in the customs union after Britain leaves was "not credible."

Additional reporting Michael Staines and Jack Quann