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Political leaders in London, Wales and Scotland have demanded the same Brexit concessions granted to Northern Ireland that would allow them to effectively remain in the single market.

In a remarkable political fallout, Prime Minister Theresa May faced a furious backlash from politicians in Scotland, Wales and London on Monday amid reports she had conceded Northern Ireland would keep EU trade rules after the UK leaves the bloc in March 2019.

Scotland’s First Minister Nicola Sturgeon said that if a Brexit deal can be done that “effectively” keeps Northern Ireland in the single European market, there is “surely no good practical reason” why others should not benefit from such an arrangement.

Both Scotland and Northern Ireland voted to remain part of the EU in the referendum in June 2016.

The SNP leader stated: "Right now, Ireland is powerfully demonstrating the importance of being independent when it comes to defending your vital national interests."

Follow all the latest on Brexit developments LIVE here.

London Mayor Sadiq Khan said: "Huge ramifications for London if Theresa May has conceded that it's possible for part of the UK to remain within the single market and customs union after Brexit."

The capital - which has a population of 8.8 million - voted by a margin of 59.9 percent to remain within the EU. Turnout was high by local standards at nearly 70 percent and Mr Khan campaigned to stay in the EU.

Welsh first minister Carwyn Jones called for Wales to be allowed to continue to participate in the single market if other parts of the UK could.

He said: “We cannot allow different parts of the UK to be more favourably treated than others. If one part of the UK is granted continued participation in the Single Market and Customs Union, then we fully expect to be made the same offer.”

After a working lunch on Monday, Mrs May and European Commission President Jean-Claude Juncker said that their latest round of Brexit talks had broken up without a deal but that progress had been made.

Discussions are set to resume later this week, with both declaring themselves "confident" that a solution can be found in time for a key summit of the European Council on December 14.

Britain and the European Union appeared to be moving closer to agreement on key issues, including the status of the Irish border.

Arrangements which would effectively allow Northern Ireland to remain part of the European single union would prevent the return of a “hard border” between the North and the Republic of Ireland.

While Mrs May has previously insisted the entire UK will leave the single market, European Council president Donald Tusk made clear that Brexit talks cannot move on to trade issues unless the UK can satisfy Dublin there will be no return to a strictly-controlled border in Ireland.

Politicians in Scotland, Wales and London have campaigned for Britain as a whole to stay in the EU's single market to smooth trade relations. But Mrs May has ruled that out so far, saying Britain needed the freedom to make its own rules and trade deals.

They had rallied to the cause again on Monday after Irish government sources said that Mrs May’s government had agreed to maintain EU "regulatory alignment" for Northern Ireland, which is part of the UK but shares a land border with the Republic of Ireland, an EU member state.

Ireland’s deputy premier Simon Coveney had said Dublin government's concerns over the post-Brexit border with Northern Ireland were set to be addressed fully, amid reports the UK would allow "regulatory alignment" between north and south.

Mrs May said there had been no overall agreement.

Regulatory alignment could mean both Ireland and Northern Ireland following the same rules governing trade, to ensure that goods can continue to move freely across a "soft" border with no checks.

But making an exception of Northern Ireland may make it difficult for Mrs May to argue others cannot have the same. So far, she has argued that Brexit should follow a one-size-fits-all pattern for the whole of the United Kingdom.

The DUP has said it will not accept any Brexit deal that “separates” Northern Ireland from the rest of the UK.

DUP leader Arlene Foster said the party, which props up Mrs May's Tory minority Government, would not allow "any form of regulatory divergence" from the UK.

Ms Foster made clear the DUP would oppose the deal if it meant the effective drawing of a new border in the Irish Sea between Northern Ireland and the rest of the UK if the Westminster Government decides it wants to diverge from EU rules.

The PM broke off from negotiations with Mr Juncker for urgent telephone talks with the DUP leader after her statement.

Earlier, Mrs May's official spokesman said any agreement on the Irish border would protect the "territorial and economic integrity" of the UK.

Ahead of the meetings in Brussels, he told a regular Westminster briefing: "The PM has been clear that the UK is leaving the European Union as a whole and the territorial and economic integrity of the United Kingdom will be protected."

Responding directly to questions about the report of agreement on regulatory alignment by Irish broadcaster RTE, he said: "RTE also reported this morning we were holding a Cabinet meeting and I missed that if it occurred."