Cabinet Office minister David Lidington, the Prime Minister's de facto number two, said her high-profile Friday speech was an "ambitious opening bid" for negotiations on trade.
Now the International Trade Secretary has come forward and said Labour's stance would result in Britain being "rule takers with no say in how they were made".
After increasingly impatient appeals from European Union leaders for more detail about Britain's aims, Mrs May used her keenly awaited address at the Mansion House to put more flesh on the bones of her vision for "the broadest and deepest possible partnership" after Brexit.
"The reality is that we all need to face up to some hard facts".
"But the question is 'who decides?' and at the moment the UK Government's proposition is that even in areas of devolved competence - agriculture, the environment, fishing, justice - they should be able to impose frameworks on Scotland and Wales, and our position is that where there are devolved areas it should only be by the consent of the Scottish and Welsh parliaments, so I hope we can reach agreement".
Mr Coveney said if agreement can not be reached during tripartite talks between the UK, Ireland and the EU Commission, the backstop plan of full British alignment with customs union and single market rules that Mrs May "committed clearly" to in December's conclusion of withdrawal negotiations would have to be put in place.
As an Arctic blast swept Britain, the U.K. Prime Minister sent her warmest words yet to Brexit negotiators in Brussels.
Labour's shadow chancellor John McDonnell told Sky News' Sunday with Paterson show that the PM's Brexit speech was more about trying securing a "limited truce" within the her own party than "the interests of the country".
Teacher arrested after firing a gun in a US high school
There they found a teacher had barricaded himself in a classroom. "At the time , there is a barricaded subject in a classroom". Two of his friends offered to "take care of her" and he said he agreed, according to a 2016 police report.
In a speech Friday aimed at answering critics who accuse Britain of failing to grasp the tough realities of leaving the EU, May will call for "the broadest and deepest possible agreement - covering more sectors and cooperating more fully than any free trade agreement anywhere in the world today".
The Tanaiste says the British Prime Minister has offered no more detail than already heard about how to avoid a hard border after Brexit.
The speech was well received by her confidence and supply partners the DUP, with the party's leader Arlene Foster "particularly welcoming" a commitment to "strengthen" the United Kingdom union.
But pro-Brexit cabinet ministers have pressured her to drop May's promise that there would be "binding commitments" to mirror European Union rules in some sectors, The Times reported.
But she sought to reassure companies that an alternative system would be set up.
The Tánaiste has said he is "not sure" that the European Union will be able to support the British Prime Minister's plan for avoiding a hard border in Ireland. Later in the speech she did speak about the use of technology at the border and a customs arrangement (but not a customs union).
Labour leader Jeremy Corbyn accused Mrs May of failing to bring "real clarity", stressed he was anxious about her admittance that access to European markets would be reduced, and said she "failed to provide solutions" in particular on the Irish border.
She dismissed the commission's insistence that it would not allow "cherry-picking" of rights and obligations, insisting that this was a part of any trade negotiation.
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