Just six months before Britain is due to leave the European Union, the two sides differ on their view of the talks - the bloc says a withdrawal deal is within reach, while British officials say "big issues" still lie in the way of any agreement.
Steve Baker, Conservative MP for Wycombe, said he believed "at least 40" of his colleagues would vote against a deal along the lines the PM has proposed which would include this "backstop" for Northern Ireland.
The DUP has a confidence and supply arrangement with the government which sees its 10 MPs help it see bills through the legislative process.
Repeating his call for Mrs May to ditch the plan agreed at her country residence in July, he said: "In the referendum both sides said Leave meant leaving the customs union and single market".
By withdrawing its support, the DUP could make it hard for May to pass legislation through parliament, including the budget which will be voted on later this month.
"Why would we need checks between GB and Northern Ireland or between Northern Ireland and GB if we were an integral part of the single market of the United Kingdom?" she said.
According to spokesman Margaritis Schinas, Barnier "recalled that decisive progress must be made in time for the October European Council next week and negotiations at technical level will continue this week".
The DUP has ten MPs, although Ian Paisley will not be able to vote until 20 November after being suspended for failing to declare two family holidays paid for by the Sri Lankan government.
Mr Coveney said that the Dublin Government had various methods of staying in contact with the DUP in a very regular way.
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.
"She is being forced by the European Union".
Barnier said customs forms could be completed in advance online and the "only visible systematic checks" would involve scanning bar codes on lorries or containers on ferries or in ports.
Negotiators in Brussels remain tight-lipped, but it sounds like a deal on the withdrawal agreement is close.