The Cabinet are still examining ideas to solve the thorny Irish border issue after Britain quits the EU with no clear way forward so far.
One idea proposed by Brexit Secretary David Davis would see Northern Ireland covered by a joint regime of UK and EU customs regulations, allowing it to trade freely with both, plus a 10-mile wide “special economic zone” on the border with Ireland.
The idea has attracted scorn from all sides and has already been dismissed by Downing Street.
And Ms Foster warned customs parity with Britain after Brexit was vital for her party, whose 10 MPs support the Conservatives in Westminster under a “supply and demand” arrangement.
She told Sky News: “For us, our only red line is that we are not treated any different from the rest of the United Kingdom, that there are no trade barriers put up between Northern Ireland and our biggest market which, of course, is Great Britain.
“That’s what we will judge all of the propositions that are brought forward, we will judge it against that red line and she’s very much aware of that.
“I have confidence that she knows that she cannot bring forward anything that will breach that red line or we simply will not be able to support them.”
Last month, Cabinet ministers analysed two main options for the Irish border, a “customs partnership” proposal that would see Britain continue to collect tariffs on behalf of the EU and a technology-based “maximum facilitation” – or “max fac” – solution.
Mr Davis’ idea was dubbed “max fac 2”.
But Brussels has already rejected both schemes, with chief Brexit negotiator Michel Barnier saying on Friday that neither was “operational or acceptable”.
EU leaders including Irish Taoiseach Leo Varadkar have called for progress by the time the European Council meets at the end of June.
And Tanaiste Simon Coveney told the Irish Times yesterday the UK must produce written proposals for the border within the next two weeks.
Meanwhile, ministers today attempted to dismiss reports civil servants have been drawing up scenarios for a “Doomsday Brexit” that would leave the country short of medicine, fuel and food.
The Sunday Times said models for mild, severe and “Armageddon” reactions to no-deal exits were created, with a source saying that even the severe scenario saw the Port of Dover “collapse on day one”.
Home Secretary Sajid Javid told the BBC’s Andrew Marr Show: “I have to say I don’t recognise any bit of that at all and as Home Secretary.
“I am deeply involved in ‘no deal’ preparations as much as I am in getting a deal – I’m confident we will get a deal.
“From the work that I have seen and the analysis that has been done, those outcomes.
“I don’t think any of them would come to pass.”
He added that the Government was making progress with Brexit plans, saying: “I’m confident that as we get to the June council meeting the Prime Minister will have a good set of proposals and our colleagues in Europe will respond positively.”