The DUP has accused the PM of breaking promises over plans to avoid a hard border between Northern Ireland and the Republic of Ireland post-Brexit.
The party reacted angrily to a letter from Theresa May to its leader Arlene Foster, leaked to the Times.
Mrs Foster claimed the PM was "wedded to a border down the Irish Sea" as a fallback option for avoiding checks, if no free trade deal is reached in time.
Downing Street reiterated the PM's own 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.
Agreeing a backstop - a contingency plan designed to keep an open border between Northern Ireland and the Republic of Ireland until the UK and EU 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".
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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 letter sets out the prime minister's commitment "to never accepting any circumstances in which the UK is divided into two customs territories.
"The government will not agree anything that brings about a hard border on the island of Ireland," the spokesman added.
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".
Mrs Foster told the Times: "It appears the prime minister is wedded to the idea of a border down the Irish Sea with Northern Ireland in the EU single market regulatory regime."
In her letter to the DUP, Mrs May writes that she could not accept any circumstances in which that contingency plan "could come in to force".
But 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 was pressed on whether the Brexit withdrawal agreement would include the EU proposal to keep Northern Ireland tied to its customs union and single market.
He did not directly rule it out but told the BBC the government "will not do anything that puts our union at risk" and 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."
Former attorney general and pro-Remain Tory MP Dominic Grieve told the BBC: "I don't think the prime minister is wedded to a border down the Irish Sea, but I think the prime minister is unable to give reassurance to the DUP that such a thing could not happen."