The DUP has said Boris Johnson rejected the idea of a Northern Ireland-only backstop during a meeting at Downing Street.
DUP leader Arlene Foster and parliamentary leader Nigel Dodds met the Prime Minister on Tuesday evening.
Afterwards Mrs Foster released a statement saying Mr Johnson had again rejected the idea of a special economic zone for Northern Ireland.
“The Prime Minister rejected a Northern Ireland-only backstop in a letter to Donald Tusk on 19 August,” a statement said.
“It is undemocratic and unconstitutional and would place a tariff border between Northern Ireland and the rest of the United Kingdom. That would be unacceptable.
“During today’s meeting, the Prime Minister confirmed his rejection of the Northern Ireland-only backstop and his commitment to securing a deal which works for the entire United Kingdom as well as our neighbours in the Republic of Ireland.”
The idea of a Northern Ireland-only backstop had been reintroduced as the parliamentary impasse meant a no-deal Brexit became increasingly likely, and no agreed solution on how to manage the Irish border has yet been offered during negotiations between the EU and UK.
The Irish premier has not ruled out the idea, and Irish opposition leader Micheal Martin said on Monday that he felt it was a viable solution.
The DUP has been against the idea since its inception, labelling it “anti-democratic and unconstitutional”.
On Tuesday, Mrs Foster, whose party backed Brexit, said a deal is possible without Northern Ireland being subject to different customs regulations to other parts of the UK.
“History teaches us that any deal relating to Northern Ireland which cannot command cross-community support is doomed to failure,” she added.
“That is why the Northern Ireland backstop is flawed. Not one single unionist MLA in the Northern Ireland Assembly supports it.
“We want to see the referendum result implemented. Those blocking Brexit are causing uncertainty but more worrying they are damaging democracy by ignoring the United Kingdom’s decision.
“A sensible deal between the United Kingdom and European Union, which respects the economic and constitutional integrity of the United Kingdom, is the best way forward for everyone.”