• They also claimed she said a deal would never be possible unless Northern Ireland stayed in a customs union.
• Boris Johnson spoke to the German chancellor earlier about the proposals he put forward to the EU - but the source said she made clear a deal based on them was "overwhelmingly unlikely".
A No 10 source says a Brexit deal is "essentially impossible" after a call between the PM and Angela Merkel.
Boris Johnson spoke to the German chancellor earlier about the proposals he put forward to the EU - but the source said she made clear a deal based on them was "overwhelmingly unlikely".
They also claimed she said a deal would never be possible unless Northern Ireland stayed in a customs union.
But Labour called it a "cynical attempt to sabotage the negotiations".
Shadow Brexit secretary Sir Keir Starmer said Mr Johnson "will never take responsibility for his own failure to put forward a credible deal", and called on Parliament to "unite prevent this reckless government crashing us out of the EU".
The BBC's Brussels correspondent, Adam Fleming, said the German chancellery had not released their interpretation of the call, and it would be important to "compare and contrast".
But he said an EU official close to the negotiations in Brussels had told him Mrs Merkel's reported comments did not reflect the EU's agreed position, adding: "This is not our language."
Mr Johnson sent new proposals to Brussels last week, with the key focus being on replacing the so-called backstop - the policy negotiated by Theresa May and the EU to prevent a hard border returning to the island of Ireland - which has long been a sticking point.
After presenting the plans, government sources hoped the UK might be able to enter an intense 10-day period of negotiations almost immediately, with the aim of coming to a final agreement at an EU summit on 17 October.
But after the phone call, the No 10 source said it had been a "clarifying moment", adding: "Talks in Brussels are close to breaking down, despite the fact that the UK has moved a long way."
They said the UK was not willing to move away from the principle of providing a consent mechanism for Northern Ireland or the plan for leaving the customs union, and if the EU did not accept those principles, "that will be that" and the plan moving forward would be an "obstructive" strategy towards Brussels.
They also accused the EU of being "willing to torpedo the Good Friday agreement" - the peace process agreed in Northern Ireland in the 1990s by refusing to accept Mr Johnson's proposals.
The government argues allowing Stormont to approve part of the PM's plan is key to respecting the so-called "principle of consent" in the Good Friday Agreement, but Irish Prime Minister Leo Varadkar has warned it could actually undermine that principle by giving one party in Northern Ireland a veto over what happens to the country as a whole.
'Scupper a delay'
Mr Johnson has insisted the UK will leave the EU on the Brexit deadline of 31 October, with or without a deal. That is despite legislation passed by MPs last month known, as the Benn Act, which requires Mr Johnson to write to the EU requesting a further delay if no deal is signed off by Parliament by 19 October - unless MPs agree to a no-deal Brexit.
On Monday night, the Spectator published texts from a Downing Street source, who claimed if the deal "dies in the next few days, then it won't be revived".
The government has not denied the briefing, which also said Mr Johnson "will do all sorts of things to scupper a delay" to leaving the EU.
Earlier, the BBC's political editor Laura Kuenssberg said there was a growing expectation in government Brexit talks would fail before the week was out.
EU diplomats have told Adam Fleming there has been "zero progress" in the talks and in some areas they believe things are going backwards.
What are the PM's border plans?
Under Mr Johnson's proposals, which he calls a "broad landing zone" for a new deal with the EU:
- Northern Ireland would leave the EU's customs union alongside the rest of the UK, at the start of 2021
- But Northern Ireland would continue to apply EU legislation relating to agricultural and other products, if the Northern Ireland Assembly approves
- This arrangement could, in theory, continue indefinitely, but the consent of Northern Ireland's politicians would have to be sought every four years
- Customs checks on goods traded between the UK and EU would be "decentralised", with paperwork submitted electronically and only a "very small number" of physical checks
- These checks should take place away from the border itself, at business premises or at "other points in the supply chain"
What about the EU?
After receiving the proposals, the EU pledged to examine them carefully.
But a number of senior figures, including Mr Varadkar, warned the proposals did not form the basis for deeper negotiations - even if they believed a deal could still be done.
French President Emanuel Macron said the EU would decide at the end of the week whether a new deal was possible.
But a leaked presentation to EU diplomats revealed they were unwilling to accept the UK's plans committing to no checks on either side of the Irish border if the Northern Ireland Assembly - Stormont - is granted a veto and there is no guarantee of checks on the UK side.
BBC Brussels correspondent Adam Fleming said EU negotiators were "so nonplussed by the proposal they asked if it was a mistake".
It is understood the UK also wants continuing access to several EU trade databases, even if Stormont withholds its consent for the new arrangements.
Timeline: What's happening ahead of the Brexit deadline?
Tuesday 8 October - Last working day in the House of Commons before it is due to be prorogued - suspended - ahead of a Queen's Speech to begin a new parliamentary session.
Monday 14 October - The Commons is due to return, and the government will use the Queen's Speech to set out its legislative agenda. The speech will then be debated by MPs throughout the week.
Thursday 17 October - Crucial two-day summit of EU leaders begins in Brussels. This is the last such meeting currently scheduled before the Brexit deadline.
Saturday 19 October - Date by which the PM must ask the EU for another delay to Brexit under the Benn Act, if no Brexit deal has been approved by Parliament and they have not agreed to the UK leaving with no-deal.
Thursday 31 October - Date by which the UK is due to leave the EU, with or without a withdrawal agreement.