The Conservatives will lose the next election if Theresa May doesn't change her Brexit policy, ex-cabinet minister David Davis has warned in a stinging letter to Tory MPs.
Barnier is attempting to negotiate such a deal, but time is running out and European Union leaders have made it clear that they expect Britain's Prime Minister Theresa May to come up with a way to unblock talks before next week.
The Prime Minister said: "I would hope that everybody across this whole House will put the national interest first".
EU commission spokesman Margaritis Schinas said Mr Barnier had told officials at a private meeting earlier that "decisive progress must be made in time for the October European Council next week".
The party's 10 MPs have propped up the Tory government since May lost her Commons majority in the snap general election previous year.
But Foster refused to consider such a compromise on Tuesday, saying: "What we said to Barnier is checks of themselves are symptomatic of something different, so we only need checks if Northern Ireland is following a different regulatory regime to the rest of the UK".
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.
But the DUP could abandon this deal if Brexit means new barriers between Northern Ireland and Great Britain, Newsnight's Nick Watt said. If they jump the gun at the Budget before the Withdrawal Bill votes later, they're playing very hard ball indeed - and before they can know exactly what's going on in Brussels.
Boris Johnson has accused Theresa May of negotiating a "backstop" solution for Northern Ireland that will turn Britain into a "permanent EU colony" after Michel Barnier insisted the United Kingdom will have to stay in the customs union.
A Brexit sign is seen between Donegal in the Republic of Ireland and Londonderry in Northern Ireland at the border village of Muff, Ireland, February 1, 2018.
Losing the party's support in the Commons would mean possible defeat on the budget and a no-confidence vote.
Steve Baker, Conservative MP for Wycombe, said he believed "at least 40" of his colleagues would vote against a deal along the lines the prime minister has proposed which would include this "backstop" for Northern Ireland.
Foster stressed that she wants to reach a Brexit deal and she is staying in Brussels for the next two days in order to try and find a solution that would be acceptable to her.
The prime minister's spokesperson said government officials are in "intensive" talks with European Union negotiators in order to resolve the Irish border issue, which is the last major outstanding issue in the draft Withdrawal Agreement.
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