The DUP leader has refuted in the Times reports that the party is ready to follow some European rules after Brexit.
The newspaper said the DUP had agreed to "move its red lines" as part of a new deal to replace the backstop.
She added that the party had declared privately that she would drop her objections to regulatory checks in the Irish Sea.
Arlene Foster tweeted "Britain must go as a nation" and "anonymous sources lead to nonsense stories".
Ms Foster said the party will "not support agreements that are an obstacle to East-West trade."
DUP Brexit spokesman Sammy Wilson told BBC's Good Morning Ulster that the story "violates everything that has been said in recent days" and dismisses it as "bad journalism."
The backbone is the insurance that avoids a hard border between Northern Ireland and the Republic of Ireland after Brexit, unless and until another solution is found.
Westminster MPs rejected the resignation and resignation agreement negotiated by ex-Prime Minister Theresa May, which would have held the whole of Britain in a "temporary customs territory" with the EU, and Northern Ireland would continue to obey other EU rules.
The Democratic Unionists (DUP) had argued that they would create a border along the Irish Sea and risk the future of the Union.
On Thursday, the head of the European Parliament said he was ready to reconsider the suggestion of a back-stop just for Northern Ireland to overcome the stalemate on Brexit.
But on Wednesday, Prime Minister Boris Johnson ruled out an NI-only backstop.
Britain will leave the EU on 31 October without an agreement, unless both sides can reach a compromise.