Anonymous sources told The Times that the party would be willing to see Northern Ireland abide by some EU rules after Brexit in a deal to replace the Irish backstop.
The paper claimed the biggest party in Northern Ireland had privately said it would drop its objection to regulatory checks in the Irish Sea, something it had previously said was unacceptable since it would separate Northern Ireland politically and economically from the mainland.
But DUP leader Arlene Foster has dismissed the claims as "nonsense".
She tweeted late last night: “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,” she added.
The party has repeatedly insisted regulatory checks in the Irish Sea would be unacceptable, as they would separate Northern Ireland “politically and economically” from the mainland.
But The Times reported that the DUP had privately agreed to drop its objection to such checks.
Citing unidentified sources, The Times wrote that in return for such concessions, Brussels would abandon its insistence on Northern Ireland remaining in a customs union with the EU.
Instead, it would agree to fast-track “alternative arrangements” to ensure that the right level of duty was paid on exports without the need for a hard border.
However, when asked about the story on BBC NI's The View last night, South Belfast politician Christopher Stalford said he did not "recognise that as our party position".
Shortly after, Ms Foster took to Twitter to dispel any uncertainty over the matter.
DUP Brexit spokesman Sammy Wilson further reinforced her points on Friday morning, but in a more positive turn, also pointed to a different tone in talks between London and Dublin.
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He told BBC Radio 4's Today programme: "I think that there was a different attitude in the talks between the Prime Minister and the Taoiseach at the beginning of the week and there seemed to be less rhetoric at those discussions from what there had been in the past.
"And I suppose that's progress."
Meanwhile, Mr Johnson was warned against breaking the law over Brexit by Speaker John Bercow, who vowed "creativity" in Parliament would prevent a no-deal exit.
The warning came as the EU's chief negotiator said there is "no reason to be optimistic" that a new agreement can be brokered before the Prime Minister's deadline to ask for a delay.
Michel Barnier told political leaders in the European Parliament on Thursday that he was unable to say whether discussions with the UK Government would result in a deal by mid-October.
The PM is legally bound to ask Brussels for an extension to Article 50 if he cannot get MPs to back a deal by October 19, after Parliament approved a law designed to prevent a no deal.
Mr Barnier's warning came after the PM was forced to deny lying to the Queen in order to secure his five-week suspension to Parliament as the Halloween departure deadline looms.
Outgoing Commons Speaker Mr Bercow said in a London speech that the so-called Benn Act enforcing the extension means the only possible Brexit outcome is one approved by Parliament.
The former Tory warned that it is "astonishing" that anyone has entertained the idea that the PM could disobey the law, after Mr Johnson said he would rather be "dead in a ditch" than ask for a delay.
Mr Bercow compared refusing to ask for a delay in "what one might regard as the noble end" of Brexit to a bank robber insisting they would give their loot to charity.
If the Government comes close to disobeying the Act, the Speaker said Parliament "would want to cut off such a possibility and do so forcefully".
"If that demands additional procedural creativity in order to come to pass, it is a racing certainty that this will happen, and that neither the limitations of the existing rule book nor the ticking of the clock will stop it doing so," he said.
