First Minister Nicola Sturgeon has said Scotland should be allowed to "effectively stay in the single market" following reports that Northern Ireland and the Republic could maintain regulatory alignment after Brexit.
A leaked draft of plans for the Irish border after Brexit showed British and Irish officials had agreed proposals that would effectively keep Northern Ireland in both the single market and customs union after Brexit by retaining EU regulations - unlike the rest of the UK.
The SNP leader seized on the news as a sign that Scotland could also remain in the EU trade agreements after Britain leaves the bloc, something the pro-independence party has been pushing for since the referendum.
Her calls were echoed by Welsh First Minister Carwyn Jones and London mayor Sadiq Khan, who both demanded bespoke Brexit deals.
News of the draft document broke as Theresa May attended a working lunch in Brussels with Jean Claude Juncker, the European Commission president, as she attempted to shift Brexit negotiations to the next phase where the future trade relationship can be discussed.
Ms Sturgeon posted on Twitter: "If one part of UK can retain regulatory alignment with EU and effectively stay in the single market (which is the right solution for Northern Ireland) there is surely no good practical reason why others can't."
The UK Government has repeatedly ruled out allowing a bespoke Brexit deal for Scotland, which voted in favour of remaining in the European Union, but European Council president Donald Tusk has also made it clear that Brexit talks cannot move forward unless Dublin is satisfied that there will be no return to a hard border in Ireland.
The London mayor tweeted: "Huge ramifications for London if Theresa May has conceded that it's possible for part of the UK to remain within the single market & customs union after Brexit.
"Londoners overwhelmingly voted to remain in the EU and a similar deal here could protect tens of thousands of jobs."
And Mr Jones, the Welsh Labour leader, 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 & Customs Union, then we fully expect to be made the same offer."
However the announcement prompted anger from Ms May's allies in the Democratic Unionist Party (DUP) - who effectively prop up her minority Government - as the party wants Northern Ireland to leave the EU on the same terms as the rest of the UK.
DUP leader Arlene Foster gave an emergency press conference in Belfast where she ruled out accepting any move “which separates Northern Ireland economically or politically from the rest of the United Kingdom”.
She said: "We note the speculation emanating from the European Union exit talks regarding the Republic of Ireland and United Kingdom border.
"We have been very clear. Northern Ireland must leave the EU on the same terms as the rest of the United Kingdom.
"We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the United Kingdom.
"The economic and constitutional integrity of the United Kingdom will not be compromised in any way."
The question of how to maintain a soft Irish border has become a key issue in the Brexit talks, which also hinge on progress over the so-called divorce bill and citizens rights after the UK leaves the bloc.