Former Brexit secretary David Davis warned on Tuesday of "dire" consequences for Conservatives at the next general election if the Government sticks to its negotiating stance on European Union withdrawal.
"If even unionists in Northern Ireland care less about the territorial integrity of the United Kingdom than pursuing Brexit, then it really raises questions about the type of union we're in, and indeed what unionism means", she said.
Remaining in a customs union would prevent Britain being able to make its own trade deals but it was the only way to prevent customs and regulatory checks being carried out between mainland Britain and Northern Ireland, he said.
Mrs May told her Cabinet that Britain will not accept an European Union withdrawal deal without a "precise" political declaration setting out how its requirements on trade and security will be delivered.
"We are still open - we are still open - to the idea of".
Yesterday, in what appeared to be an authorised briefing operation, senior DUP sources nearly simultaneously told the BBC and Sky News that it was prepared to vote down the budget.
It would damage the economic well being of Northern Ireland.
The ten Unionist MPs prop up Theresa May's government, which has no majority, as part of a confidence and supply deal in which they are relied on to vote through the government's Budgets.More news: Category 4 Hurricane Michael being felt on Florida coast
Writing in the Telegraph, Sammy Wilson MP, the DUP's Brexit spokesman, said Mrs May was pursuing "the road to parliamentary defeat" because his party would vote against any deal that included the proposed backstop.
One complicating factor is that keeping Northern Ireland in the single market would have to be agreed by the Stormont Assembly, which has not met since collapsing in January 2017.
The backstop will apply if the Irish border can not be kept frictionless in the context of a wider deal.
Speaking alongside DUP MEP Diane Dodds, Mrs Foster said she had not yet seen a fresh text on the Irish backstop, a guarantee of no hard border on the island of Ireland, from the British government.
Among those due to attend today's War Cabinet are: the Prime Minister's effective deputy David Lidington; Jeremy Hunt, the Foreign Secretary; Sajid Javid, the Home Secretary; Liam Fox, the International Trade Secretary; Greg Clark, the Business Secretary; Michael Gove, the Environment Secretary, and Gavin Williamson, the Defence Secretary.
Earlier Mrs May urged MPs to "put the national interest first" when deciding whether to approve the Brexit deal in a crunch vote likely later this year.
Some Brexiteers say those proposals would ensure the EU kept control over swathes of the British economy and thus run counter to the spirit of her manifesto pledge to leave the EU Customs Union and the Single Market.
He added: "Agreement is within reach if we have the negotiations on October 17 at the next council meeting".