The DUP leader has rejected reports the party is prepared to abide by some European rules after Brexit.
The unionist party had agreed "to shift its red lines" as part of a new deal to replace the backstop, said The Times newspaper, citing unidentified sources.
It added the party had said privately that it would drop its objections to regulatory checks in the Irish Sea.
Arlene Foster tweeted the "UK must leave as one nation" and "anonymous sources lead to nonsense stories".
What is the DUP's position?
The DUP previously said it wanted a "sensible deal" but would not support any arrangement that could see Northern Ireland treated differently than the rest of the UK.
UK must leave as one nation. We are keen to see a sensible deal but not one that divides the internal market of the UK. We will not support any arrangements that create a barrier to East West trade. Anonymous sources lead to nonsense stories. #frontpages— Arlene Foster (@DUPleader) September 12, 2019
The party's Brexit spokesperson Sammy Wilson dismissed the story as "bad journalism", saying it went against "all of what has been said in recent days".
The DUP had already said it might accept certain European regulations that were "essential to the operation of industry in Northern Ireland and which don't impact with our relationship with our main market in Great Britain" - if those were considered by the Northern Ireland Assembly.
"We're a unionist party - we're not going vote for any arrangement which makes us different than the rest of the United Kingdom and forms a border between Northern Ireland and the rest of the United Kingdom," said Mr Wilson on Friday.
What do 'checks in the Irish Sea' mean?
There would be a need for checks at the Irish Sea if Northern Ireland remained aligned to some EU rules but the rest of the UK did not - the so-called "Northern Ireland-only backstop".
These could take two forms:
- Customs - to make sure the right EU tariffs have been paid
- Regulatory - to make sure goods meet EU safety and quality standards
The government has been adamant it would never accept Irish Sea customs checks.
But it has also been careful not to completely close down the prospect of regulatory checks.
Mrs Foster has said the DUP would not support a Brexit deal that could lead to any new checks on goods coming into Northern Ireland from elsewhere in the UK.
However, the party has previously suggested it could be open to the idea of an all-Ireland food standards zone, which would require certain checks at ports and airports in Northern Ireland.
How is that different from the backstop?
The backstop is the insurance policy to avoid a hard border between Northern Ireland and the Republic of Ireland after Brexit, unless and until another solution is found.
Early in the Brexit negotiations the idea of a backstop that only applied to Northern Ireland was suggested.
That would involve Northern Ireland alone remaining in the EU's single market and customs union, leaving the rest of the UK (England, Scotland and Wales) free to strike trade deals.
But the DUP - which at the time propped up Theresa May's minority Conservative government - objected to that.
It said it would not accept any additional Northern Ireland-only checks because it was concerned any differences between Northern Ireland and Great Britain could threaten the union.
Instead, the idea of a backstop covering the whole UK was adopted.
This would involve the UK retaining a very close relationship with the EU for an indefinite period, keeping all of the UK in a "temporary customs territory" with the EU and seeing Northern Ireland also continuing to follow other EU rules.
That means that goods coming into Northern Ireland from elsewhere in the UK would need to be checked to see if they meet EU standards.
Those arrangements would apply unless and until both the EU and UK agree they were no longer necessary.
The backstop would not apply if the UK left the EU without a deal but the potential problems with the border would remain.
But MPs rejected that backstop when they voted down the withdrawal agreement negotiated by Mrs May.
The DUP opposed it, saying it would create a border down the Irish Sea and risk the future of the union.
On Thursday, the head of the European Parliament said he was willing to revisit the proposal of a Northern Ireland-only backstop to break the Brexit deadlock.
What about an all-Ireland food zone?
Prime Minister Boris Johnson has said he is committed to getting rid of the backstop, describing it as divisive and anti-democratic.
The UK government has floated the idea of a single zone on the island of Ireland for food standards.
The EU has a strict rule that products from a non-member state must be checked at the point of entry.
And many trade experts suggest the only way to prevent those checks at the Irish border would be for the two parts of the island to have the same standards.
In effect, that would mean Northern Ireland would have to continue to follow EU standards.
And that would mean some food products coming from elsewhere in the UK would be subject to new checks and controls at Northern Ireland ports.
In fact, the island of Ireland is already a single zone for animal health, which means all livestock coming into Northern Ireland from Great Britain is checked on entry.
But the Irish government is sceptical about the proposal, with the Taoiseach (prime minister) Leo Varadkar saying it was not enough on its own.
"We would need a single Irish economic zone, or whatever you would like to call it, to cover more than agriculture and food," he said.
On Friday, Mr Varadkar said the gap between the UK and the European Union in Brexit talks was "very wide" and that British proposals for the Irish border fell "very far short" of what was needed.
What influence does the DUP have at Westminster?
The DUP had been propping up the Conservatives in a confidence-and-supply pact since June 2017, with the votes of its 10 MPs giving the government a majority to get legislation passed in Parliament.
However, last week 21 Conservative rebels had the whip removed for voting against the party on Brexit legislation; that meant the DUP votes were no longer enough to give the government a majority.
Mrs Foster has denied her party's influence with the prime minister has waned as a result.
After she and other DUP figures held talks with the prime minister on Wednesday, Boris Johnson ruled out an NI-only backstop.
Mr Wilson said that if Mr Johnson changed his mind "he has still got the arithmetic of the House of Commons to get through".
He added: "Aside from us, there are a sizeable group within his party who will not sign up to the break-up of the UK or the tearing up of the Belfast Agreement, which is effectively what a backstop will do."
The UK is due to leave the EU on 31 October without a deal unless both sides can reach a compromise.
Last week, MPs passed a bill to force the prime minister to ask for an extension beyond the 31 October Brexit deadline if a deal is not reached with the EU, but this does not completely rule out a no-deal Brexit.