DUP leader Arlene Foster has said her party will "not be able to support" Theresa May's latest proposals aimed at resolving the Brexit deadlock.
The party accused the PM of breaking promises over plans to avoid a hard border between Northern Ireland and the Republic of Ireland.
The plans were revealed in a letter from Theresa May to Mrs Foster, leaked to the Times.
Downing Street reiterated the PM's commitment to avoiding a hard border.
Mrs May relies on the support of the DUP's 10 MPs in key votes because she does not have a majority in the House of Commons.
PM 'needs to reflect'
Agreeing a backstop - a contingency plan designed to keep an open border between Northern Ireland and the Republic of Ireland until the UK and European Union settle their future relationship - remains the main obstacle in the way of a wider deal between the two sides.
Mrs May describes it in her letter to the DUP as "an insurance policy that no-one in the UK or the EU wants or expects to use".
The DUP leader said the party could not back the deal if it came before Parliament and said the PM "needs to reflect" on that.
Mrs Foster told the BBC that a response to a letter sent to the PM last week "unfortunately does say that she's in a position where she's considering regulatory alignment".
"We would not be able to support this if it came to parliament in the form that it is in the letter," said the DUP leader.
"Now there are stages to go through before it comes to parliament. She still has to have a cabinet meeting in relation to this matter and we believe that there is a chance for her to reflect on the fact that we will not be able to support it in its current form.
Mrs Foster added: "I believe that not only would we not be able to support what she has sent to us but that there are many others that wouldn't be able to support it in her own party as well."
If sufficient progress is made on the issue in the next few days, it is thought a special cabinet meeting could be held early next week for ministers to approve the draft agreement on the terms of the UK's exit.
A Downing Street spokesman said: "The government will not agree anything that brings about a hard border on the island of Ireland."
What is the row about?
The UK and EU have failed, so far, to reach agreement on how to ensure there are no border checks between Northern Ireland and the Republic of Ireland, if a free trade deal is not in place by the end of the post-Brexit "transition period" in December 2020. This is known as the "backstop".
The EU's proposal, to keep Northern Ireland in its customs union and single market, is unacceptable to the UK because it would mean Northern Ireland having different regulations to the rest of the UK.
But the UK's proposal, which would effectively keep the whole of the UK in the EU customs union for a limited period after Brexit, includes an "expected" end date of 2021 - something which is unacceptable to the EU, which says any "backstop" must apply "unless and until" it is no longer needed.
It says its Northern Ireland-only proposal should remain in place, in case the issue of the Irish border was not sorted out by that end date.
What has annoyed the DUP?
The BBC's John Campbell said what was upsetting the DUP in Mrs May's letter was the issue of single market regulations, not customs.
He said the letter suggested the unique circumstances of Northern Ireland may require alignment with EU single market regulations "in some scenarios" adding that any such alignment would have to be "carefully circumscribed to what is strictly necessary to avoid any hard border".
The DUP has interpreted the wording of her letter to mean that the EU's proposal will be in the Brexit divorce deal, despite Mrs May's insistence it will never come into effect.
BBC political correspondent Iain Watson said the row came down to an issue of trust between Mrs May and her DUP allies, who were suspicious she might sign up to a deal with the EU they did not agree with.
What are other politicians saying about the row?
Chancellor Philip Hammond said: "We have always said that we can't accept the [EU] commission proposal for a Northern Ireland-specific solution."
Irish PM Leo Varadkar told a press conference: "The most important thing to me is the objective, and that is to give everyone in Northern Ireland the assurance that a hard border will not develop between north and south, no matter what else may happen in the years ahead.
"That is why we are seeking one that is legally operative and one that gives us that guarantee that is necessary."