Theresa May’s proposed new economic partnership with the EU “will not work”, the head of the European Council has said.

Donald Tusk said the plans risked undermining the EU’s single market.

He was speaking at the end of an EU summit in Salzburg where leaders of the 27 remaining member states discussed Brexit.

Mrs May said her proposals were the “only serious credible” way to avoid a hard border in Northern Ireland.

She said she had held “frank” talks with Mr Tusk, adding: “Yes, concerns have been raised and I want to know what those concerns are.”

There was “a lot of hard work to be done”, she said, but added that the UK was also making preparations in case no deal could be reached.

Mrs May reiterated that she would not accept the EU’s “backstop” plan to avoid a hard border in Northern Ireland, and said the UK would shortly be bringing forward its own proposals.

The UK is due to leave the EU on 29 March 2019, and both sides are trying to reach a deal in time.

There is still no agreement on some issues, including how to avoid new checks on the Northern Ireland-Republic of Ireland border.

At a press conference, Mr Tusk said there were some “positive elements” in the UK’s blueprint for future relations with the EU, which was agreed by ministers at Chequers in July.

But, he added: “The suggested framework for economic cooperation will not work.”

Mr Tusk added that October would be the “moment of truth” for reaching a deal, and that “if the conditions are there” an additional summit would be held in November to “formalise” it.

Analysis by BBC political editor Laura Kuenssberg

Can the prime minister really cling on to her Chequers plan now?

The EU clearly won’t accept it as it stands. Significant chunks of her party won’t wear the deal either. The opposition parties won’t back her.

Of course there are tactics at play here. One government minister has already suggested that the EU always knows how to overplay their hand.

Sometimes in negotiations, there needs to be a crisis to focus minds.

And in her press conference, Theresa May seemed frustrated that her proposals perhaps have not been properly digested or considered yet.

Of course this is only one day, one set of fraught meetings, in a tangled and lengthy process.

But as things stand, it seems Theresa May is going to have to budge, or walk away.

The EU leaders had been discussing the UK’s plans, which were presented to them by Mrs May on Wednesday evening.

German Chancellor Angela Merkel said “substantial progress” was needed on the UK’s withdrawal agreement by the next European Council meeting in October, with “still a large piece of work” on the separate issue of future trade relations with the UK.

The 27 remaining EU members were “united that, in the matter of the single market, there can be no compromises,” she said.

French President Emmanuel Macron said Brexit had been “pushed by certain people who predicted easy solutions”.

He added: “Brexit has shown us one thing – and I fully respect British sovereignty in saying this – it has demonstrated that those who said you can easily do without Europe, that it will all go very well, that it is easy and there will be lots of money, are liars.

“This is all the more true because they left the next day, so they didn’t have to manage it.”

As well the criticism from the EU, Mrs May’s proposal for the UK to sign up to a common rule book for trade in goods and a combined customs territory with the EU is unpopular with many in her own party, who believe it will erode British sovereignty and is not what people voted for when they backed Brexit in the 2016 referendum.

Former Brexit secretary David Davis said although EU leaders “were trying to give some warm words” it was now going to be “very, very difficult” to meet their requirements: “So it’s time for a reset, time for a rethink”.

And another prominent Brexiteer, Tory MP Jacob Rees-Mogg, said Mr Tusk’s remarks signalled the end for the Chequers proposals.

“I think Chequers now has no supporters at all,” he told the BBC.

“I think the time has come for Mrs May to say ‘This is not going to work’.”

For Labour, shadow Brexit secretary Sir Keir Starmer said Mrs May had to “urgently drop her reckless red lines and put forward a credible plan for Brexit”.

Earlier Scotland’s First Minister Nicola Sturgeon said Mrs May must delay Brexit beyond next March if there is not a detailed agreement on future trading arrangements.