Under the highly complex blueprint, the Northern Irish economy would come under a regime of both EU and UK rules to allow free trade with both.
Proposals for a 10-mile wide buffer along the 310-mile border were also being considered to allow farmers and other traders to continue to move unhindered.
Officials hoped the blueprint, based on the tiny principality of Lichtenstein’s relationship with the EU, would avoid the need for border checkpoints.
But the move infuriated the Democratic Unionist Party, whose 10 Westminster MPs are propping up Prime Minister Mrs May’s minority government.
DUP MP Sammy Wilson said: “It immediately raises all kinds of questions, first of all about the commitment not to have Northern Ireland treated differently from the rest of the UK.
“If it is kite-flying, then we need to cut the strings on that kite pretty quickly.”
Labour MP Virendra Sharma also rubbished the idea, saying: “With these latest proposals anyone would think the Government is making this up as they go along.”
A source at David Davis’s Department for Exiting the EU said the “dual market” blueprint was among a range of ideas being looked at.
They said: “There are lots of ideas being kicked around.
“This is just one of them. The chance of this being where we end up is slim.”
The source also distanced Mr Davis from the idea, saying the Cabinet minister was working hard to find a solution to the problem but was not wedded to one particular blueprint.
Officials floated the idea after ministers were at loggerheads over two other proposals.
Mrs May had backed the idea of a “customs partnership” with the EU where the UK would continue to collect EU tariffs on imports crossing the border into Northern Ireland.
But this has been rejected by a majority of ministers on her Cabinet Brexit sub-committee.
Foreign Secretary Boris Johnson and Environment Secretary Michael Gove were among the fiercest critics, claiming the plan would effectively keep Britain tied to Brussels for ever.
A Whitehall source yesterday suggested that plan had been dropped altogether. They told a newspaper: “It’s done. It’s taken too many blows.
“That’s not just the Brexiteer view – obviously the Brexiteers don’t like it. That’s the view now from the Cabinet Office and even Downing Street.
“Obviously there are a lot of people who still want to push ahead with it, but they know it’s unlikely to happen now. They know it’s taken too much heat.”
Another controversial plan, dubbed “Maximum Facilitation” or “Max Fac”, is still under discussion.
It involves using digital technology to monitor goods flowing over the border.