Ministers say no part of the UK will be treated differently in the Brexit talks as Labour branded their approach an “embarrassment”.
No agreement has been reached with the EU after a DUP backlash against proposals for the Irish border.
Brexit Secretary David Davis told MPs the government was close to concluding the first phase of talks.
Theresa May is due to speak on the phone to the DUP’s leader Arlene Foster to try to reach agreement.
Mrs Foster told the Republic of Ireland national broadcaster RTÉ that her party only saw the text of the deal on Monday morning, despite asking to see it for five weeks.
She said it was a “big shock”, adding: “Once we saw the text, we knew it was not going to be acceptable.”
The UK is due to leave the EU in March 2019 and Mrs May is under pressure to reach agreement on the Northern Ireland border so negotiations can move forward.
The prime minister needs the support of the DUP – the Democratic Unionist Party – which is Northern Ireland’s largest party and has 10 MPs at Westminster, because she does not have a majority to win votes in the House of Commons.
Responding to an urgent question from Labour in the Commons on Tuesday, Mr Davis defended the controversial proposal for “regulatory alignment” between Northern Ireland and the Republic of Ireland – intended to avoid the need for border checks after Brexit – saying this would apply to the whole of the UK.
The DUP is unhappy about any agreement which treats Northern Ireland differently.
It would not mean “having exactly the same rules” as the EU, Mr Davis said, but would involve “sometimes having mutually recognised rules”.
Backbencher Jacob Rees-Mogg warned that “regulatory divergence” from the EU after Brexit was a “red line”.
Labour’s Brexit spokesman Sir Keir Starmer said that when the DUP objected to the draft agreement, “fantasy met brutal reality”.
“The DUP tail is wagging the Tory dog,” he said, in a reference to the deal struck between the parties to help the Conservatives survive key votes.
Mr Starmer also called for the government to drop its plan to enshrine the 29 March 2019 Brexit date in UK law.
Meanwhile, former Tory leader Iain Duncan Smith suggested the UK should walk away from the negotiations if the EU does not change its position.
He told BBC political editor Laura Kuenssberg that Brussels needed to “back off” or the UK would “get on with other arrangements which are not going to be beneficial to you”.
Dublin – which as an EU member is part of its single market and customs union – has been calling for written guarantees that a “hard border” involving customs checks on the island of Ireland will be avoided after Brexit
It is concerned this could undermine the 1998 peace treaty – the Good Friday Agreement that brought an end to 30 years of sectarian conflict in Northern Ireland.
Mr Davis said that while the “integrity” of the single market and customs union must be respected after Brexit, it was “equally clear we must respect the integrity of the United Kingdom” and individual nations could not have separate arrangements.
Conservative MPs clashed on the way forward, with pro-Remain Anna Soubry claiming there was a Commons majority to keep a close relationship with the single market and customs union.
But former minister Owen Paterson, a Eurosceptic, said the government should be prepared to walk away from talks.
What happened in Brussels
Mrs May needs to show “sufficient progress” has been made so far on “divorce” issues before European leaders meet on 14 December to decide whether to allow talks on future trade relations to begin.
The three issues that need to be resolved are the Northern Ireland border, citizens’ rights and the amount of money the UK will pay as it leaves.
Talks between Prime Minister Theresa May and European Commission president Jean-Claude Juncker broke up without agreement on Monday, after the Democratic Unionist Party (DUP) objected to a draft agreement on the future of the Irish border.
Key to the row is how closely aligned Northern Ireland’s regulations will be with those of the Republic of Ireland, and the rest of the EU, in order to avoid a “hard” border.
DUP leader Arlene Foster was invited to hold talks with Mrs May in London on Tuesday, but the party’s Westminster leader will meet the government’s chief whip instead.
Ireland’s deputy prime minister Simon Coveney said Dublin would not budge from its position on the border.
The EU is treating the row as a “domestic British political issue”, BBC Brussels correspondent Adam Fleming said.
“The show is now in London,” said a European Commission spokesman.
Reacting to leaks of the draft agreement, Scottish First Minister Nicola Sturgeon said that if Northern Ireland was allowed to operate under different rules there was “surely no good practical reason” why other parts of the UK could not do the same and appealed to other parties to keep the UK in the EU’s customs union.
This message was echoed by Welsh First Minister Carwyn Jones and London Mayor Sadiq Khan.
Scottish Conservative leader Ruth Davidson called for a UK-wide deal, rather than specific arrangements for Northern Ireland.
Downing Street has insisted the border was not the only outstanding problem and disagreement remains over the role of the European Court of Justice in overseeing EU citizens’ rights in the UK after Brexit.
The prime minister is expected back in Brussels for further talks before the end of the week.