The whole of the UK should adopt regulatory alignment with the European Union if it is the only way to avoid a hard border on the island of Ireland, the Scottish Conservative leader has said.
Ruth Davidson has urged the Prime Minister not to give any separate Brexit deals to singular parts of the UK.
STV News understands the Scottish Conservative leader warned May in a phone call on Tuesday morning she would not accept any separate deals for singular parts of the UK.
May was close to signing off a regulatory alignment deal for Northern Ireland, which would mean the country would maintain the same regulations as the EU even if the rest of the UK diverged.
The agreement was rejected by DUP leader Arlene Foster.
Foster, whose support May needs to have a working majority in the Commons, said her party will not accept "which separates Northern Ireland economically or politically from the rest of the United Kingdom".
The Prime Minister is chairing a cabinet meeting on Tuesday morning and, according to ITV's political editor Robert Peston, will echo Davidson's call for UK-wide deal on regulations.
In a statement posted on Twitter, Davidson said: "The question on the Brexit ballot paper asked voters whether the UK should stay or leave the European Union - it did not ask if the country should be divided by different deals for home nations.
"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."
She added: "All sides agree there should be no return to the borders of the past between Northern Ireland and the Republic of Ireland.
"Similarly, jeopardising the UK's own internal market is in no one's interest.
"If regulatory alignment in a number of specific areas is the requirement for a frictionless border, then the Prime Minister should conclude this must be on a UK-wide basis."
The 13-strong group of Scottish Conservative MPs backed Davidson in a joint statement following a meeting on Tuesday afternoon.
It read: "The Scottish Conservative group met today and unanimously agreed that we fully support the comments from Ruth Davidson MSP that the terms of any Brexit deal with the EU should be UK-wide."
The First Minister and other devolved leaders called for similar deals to what was being proposed for Northern Ireland almost immediately after details of the draft agreement were leaked.
"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 First Minister tweeted.