They would be able to vote on the terms, including the backstop plan - but worldwide trade secretary Liam Fox, a Brexiteer, dismissed the idea.
And he added: "The UK tells us it wants to avoid any control".
Michel Barnier also stated that Brussels would not be "intimidated" by Brexiteers who try to blame the bloc for not securing the Brexit deal they want.
But Mr Barnier said: "Our backstop can't be extended to the whole UK. Why?"
May's ministers are split on what kind of customs deal Britain should seek from the European Union ahead of a summit of the bloc's leaders which will try to make progress on the stalemated Brexit negotiations this month. The Irish Republic is primarily interested in avoiding a hard border and is likely to accept the compromise, as Irish Foreign Minister Simon Coveney underscores that it is "vital" that a legally-binding backstop agreement is in place.
Theresa May's latest Brexit proposals received a frosty reception in Brussels, with chief European Union negotiator Michel Barnier raising objections to her plan for a time-limited customs backstop covering the whole UK.
Friends of Mr Johnson said it was disappointing that the private dinner had been covertly recorded but senior Conservative Sarah Wollaston said dressing up the comments "under the cover of a "private" discussions won't wash". "A people's vote must be held before March 29, 2019, to decide our country's future", the organization said in its roadmap.
If the future trade agreement finds it impossible to provide for a frictionless invisible border, then, "specific (technological) solutions" will apply to Northern Ireland.
A national conversation would then be launched in early 2019, with the vote being held before March 29, when the United Kingdom is now scheduled to leave the EU. We will go right to the heart of Britain and the core of the debate: "in Doncaster, Stoke, and Mansfield - places where the anguish of the people before 2016 has so clearly shaped the political landscape after it". We decided it had to be done on the basis of collective responsibility - in other words, everyone signs up to it.
After 24 hours of scrambling to keep her Brexit minister, David Davis, on board, Mrs May published what the government called the temporary customs arrangement.
Mr Barnier said he continues to have "great respect" for the prime minister.
"Several people around the table said that", the source said on condition of anonymity.
Asked if she expected Davis to remain in his position on Thursday, May's spokeswoman said: "Yes... of course".
Mr Dodds said: "As we exit the European Union we must ensure there is no new border in the Irish Sea between Northern Ireland and our greatest market in Great Britain".
The Prime Minister was "going to go into a phase where we are much more combative with Brussels", the Cabinet minister said.