David Davis insisted Northern Ireland would not be "left behind" in the single market and customs union after Brexit as Theresa May worked to get withdrawal negotiations back on track.
The British prime minister was telephoning DUP leader Arlene Foster and Sinn Féin leader Michelle O'Neill on Tuesday for urgent talks as pressure mounted on her to break the logjam.
Mrs May's crunch withdrawal talks with the EU in Brussels on Monday ended without agreement after the DUP refused to accept proposals which would have shifted Northern Ireland's customs border to the Irish Sea, in order to maintain a soft border with the Republic of Ireland.
She is now facing internal demands not to cut separate deals for different parts of the UK, after the leaders of Scotland, Wales and London called for carve-outs to remain in the single market and customs union.
Mr Davis stressed that the idea that Northern Ireland would be left inside the single market or customs union to avoid a hard border on the island of Ireland was a "falsehood".
Answering an urgent question in the House of Commons, the Brexit Secretary said: "The suggestion that we might depart the European Union but leave one part of the United Kingdom behind, still inside the single market and customs union - that is emphatically not something that the UK Government is considering.
"So when the First Minister of Wales complains about it, or the First Minister of Scotland says it's a reason to start banging the tattered drum of independence, or the Mayor of London says it justifies a hard border around the M25 - I say they're making a foolish mistake.
"No UK government would allow such a thing, let alone a Conservative and Unionist one."
Meeting Spanish Prime Minister Mariano Rajoy in Downing Street, Mrs May said: "Our talks with the European Union have made a lot of progress.
"There are still a couple of issues we need to work on, but we'll be reconvening in Brussels later this week as we look ahead to the December European Council.
"But I know in everything we do we want to ensure, particularly, that we recognise the needs of Spanish citizens here in the UK , and UK citizens living in Spain."
Mr Rajoy said the Prime Minister, who briefed Cabinet colleagues on Tuesday, was doing a "sterling job, fantastic job".
"The speech that was delivered by Mrs May in Florence was a very important speech, and I am personally convinced that the second stage of negotiations will begin soon," he said.
Mrs May plans to return to Brussels before the end of the week, with time running out to persuade leaders of the remaining 27 EU nations at a summit on December 14-15 that "sufficient progress" has been made on divorce issues to move Brexit negotiations on to their second phase on trade.
Her official spokesman rejected suggestions that the new deadline for agreement on divorce issues was Friday, insisting the British government is working towards the European Council summit.
As Mrs May grappled to find a form of words acceptable to the DUP, its leader, Arlene Foster, shared a tweet from Scottish Conservative leader Ruth Davidson insisting that any Brexit agreement must be UK-wide.
Ms Davidson said: "While I recognise the complexity of the current negotiations, no Government of the Conservative and Unionist Party should countenance any deal that compromises the political, economic or constitutional integrity of the United Kingdom."
Mr Davis said "all parties" in the negotiations were confident of progress, noting comments from senior EU figures Jean Claude Juncker and Donald Tusk, whom Mrs May met on Monday.
But the Tánaiste Simon Coveney, insisted Dublin would not budge from its position on the border, while agreeing to work on "presentational issues".
"We don't want to give the impression the Irish government is going to reverse away from the deal we felt we had in place and agreed yesterday."
Scottish First Minister Nicola Sturgeon called on Labour leader Jeremy Corbyn to "get his act together" and unite with Remain-backing Tories to "force a different, less damaging Brexit leaving the UK in the single market and customs union".
And former top Foreign Office official Lord Ricketts said the row was was "damaging" for Mrs May and will leave EU leaders with the impression that she lacks the authority to get through Brexit negotiations.
"We're used to prime ministers going to Brussels and having a row with the EU and coming back without an agreement, but to go agree with the EU and then have a row on your own side is inconvenient," he told BBC Radio 4's Today programme.
The Government's approach received a withering assessment from ex-Labour leader Ed Miliband, who tweeted: "What an absolutely ludicrous, incompetent, absurd, make it up as you go along, couldn't run a p**s up in a brewery bunch of jokers there are running the government at the most critical time in a generation for the country."
