A government source said that ministers feared they could be bounced into swallowing several potential changes to the customs arrangement and the areas of European Union law that the United Kingdom must follow after Brexit.
Earlier Mrs Foster tweeted an image from her meeting with Mr Barnier, emphasising that she will not tolerate any form of barrier between Northern Ireland and the rest of the UK.
The small Northern Irish party which props up May's government again drew what it described as the "only one red line" in its talks in Brussels on Tuesday, ruling out any checks on trade with the British mainland after Brexit. She has not resisted the unreasonable calls for Northern Ireland to be treated differently in terms of regulation, in terms of participation and future trade agreements.
Health Secretary Matt Hancock told BBC Radio 4's Today the "whole nation" should get behind Theresa May as she strove to get the best deal for the country in "difficult circumstances".
On Friday (12), the European Union ambassadors from the 27 non-British member states will meet in Luxembourg to discuss Brexit in the run-up to the summit, a diplomatic source said.
The UK Government has promised to set out further proposals on the "backstop" arrangement aimed at avoiding a hard border with Ireland, with speculation that it could involve regulatory checks on goods travelling between Britain and Northern Ireland.
Technical talks are continuing at official level in Brussels this week, but no visit by Brexit Secretary Dominic Raab has been announced, and Downing Street would say only that new proposals on the border issue would be released "in due course".
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Mboweni is a member of the ANC's national executive committee, one of the ruling party's highest decision-making bodies. The main opposition Democratic Alliance party welcomed Mboweni's appointment, saying he was a safe pair of hands.
Davis added that it was "in the party's interest and the national interest" to "reset" the UK's negotiating strategy "immediately".
"If Theresa May doesn't take our concerns on board, she may not be the leader to take us through Brexit".
But he said a "Canada-plus-plus-plus" deal was "within our grasp with political will and imagination".
In parliament, her deputy Nigel Dodds rammed the point home, warning Raab not to "repeat that mistake" from December - when the DUP nearly pulled the rug out from an interim accord negotiated by May by scuppering a proposal that would have seen regulatory alignment between the European Union and the province.
The former chancellor said that would reveal the "hardline Eurosceptic views" of the "Bennites" in the Labour leadership and the "right-wing nationalists" on the Tory benches were in the minority in the Commons.
However, May has not yet shown the new proposal to Foster, whose party the prime minister depends upon for her Commons majority.
The meeting was initially framed as the deadline for agreement on the terms of United Kingdom withdrawal from the European Union, as well as the publication of a political declaration on future relations in areas such as trade and security.
But with both sides confirming that differences remain on issues such as the status of the Irish border, expectations are growing that the final moment of decision will be put back to a special summit in November.