UK Prime Minister Theresa May has said the EU must treat the UK with more "respect" in Brexit negotiations.
In a statement at Downing Street she said for EU leaders to reject her plan with no alternative at this "late stage of negotiations" was "not acceptable".
She said talks had reached an "impasse" and could only be unblocked with "serious engagement" from the EU side.
At an EU summit on Thursday, European Council President Donald Tusk said Mrs May's plan "will not work".
The pound's fall against the dollar and the euro deepened following Mrs May's statement.
The UK is due to leave the EU on 29 March 2019.
The two sides are trying to reach a deal by November so it can be ratified in time.
They want to avoid a hard border - physical infrastructure like cameras or guard posts - between Northern Ireland and the Irish Republic but cannot agree on how.
Mrs May says her plan for the UK and EU to share a "common rulebook" for goods, but not services, is the only credible way to avoid a hard border.
She tried to sell the plan directly to EU leaders at a summit in Salzburg, Austria, this week.
Speaking afterwards, Mr Tusk said that while there were some "positive elements" in Mrs May's proposals, EU leaders had agreed that "the suggested framework for economic co-operation will not work, not least because it is undermining the single market".
On Friday, the prime minister said: "Throughout this process, I have treated the EU with nothing but respect. The UK expects the same, a good relationship at the end of this process depends on it.
"At this late stage in the negotiations, it is not acceptable to simply reject the other side's proposals without a detailed explanation and counter proposals."
She said the two sides were still "a long way apart" on two big issues: the post-Brexit economic relationship between the UK and EU, and the "backstop" for the Irish border, if there is a delay in implementing that relationship.
The two options being offered by the EU for the long-term relationship - for the UK to stay in the European Economic Area and customs union or a basic free trade agreement - were not acceptable, she added.
The first would "make a mockery of the referendum" she said, while the second would mean Northern Ireland would be "permanently separated economically from the rest of the UK by a border down the Irish Sea."
Mrs May said no UK prime minister would ever agree to that: "If the EU believe I will, they are making a fundamental mistake."
The prime minister attempted to reassure EU citizens living in the UK that, in the event no deal can be reached "your rights will be protected".
She said "no-one wants a good deal more than me" but added: "But the EU should be clear: I will not overturn the result of the referendum. Nor will I break up my country."
Mr Tusk followed up his remarks on Thursday by posting a photograph on Instagram of he and Mrs May looking at cakes with the caption: "A piece of cake, perhaps? Sorry, no cherries."
The EU has argued that the UK cannot "cherry-pick" elements from its rulebook.
That was criticised by some Conservatives, including Brexiteer Iain Duncan Smith who described it as "quite insulting".