At the same time, the Sunday Times reported that the Prime Minister was on course to agree a future economic partnership that would leave open the possibility of a Canada-style free trade deal sought by Brexiteers.
May's plans are likely to appease all wings of the Conservative Party, and could even appeal to some Labour MPs, the newspaper said.
May later reiterated to Austrian Chancellor Sebastian Kurz that she believed a withdrawal accord was 95 per cent complete, and was "confident" of a deal on the Northern Irish backstop.
A senior EU source said the United Kingdom prime minister would need determine whether she could sell an open-ended customs union to her party, but the calculation in Brussels was that it would be acceptable to the House of Commons as a whole.
According to the Sun, Chief Whip Juilan Smith has told the Prime Minister the Government could only rely on 15 Labour MPs voting in favour of a deal rather than the 40 that have been previously claimed.
May's secret concessions from the European Union centre on checks at the Northern Irish border with the Republic of Ireland, which has so far been one of the biggest sticking points in Brexit negotiations.
The Sunday Times reported that the EU would allow the creation of a whole-UK customs union that would avoid the need for a Northern Ireland border "backstop" that has been at the heart of the impasse in negotiations.
The Taoiseach also said the UK's withdrawal from the bloc had undermined the historic Good Friday Agreement that ended three decades of violence in Northern Ireland and called for a guarantee that there would be no return of a hard border with the Republic.
The bloc has also accepted that regulatory checks on goods can take place "in the market" by British officials, rather than at ports by European Union inspectors, according to the Times.
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But ministers are now insisting that the backstop agreement must contain a mechanism for the United Kingdom to terminate the backstop without the agreement of the EU.
Basically, it is an insurance policy to make sure the current frictionless border between Northern Ireland and the Republic remains open no matter what happens in negotiations between the United Kingdom and the EU.
"He recalled the prior commitments made that the backstop must apply "unless and until" alternative arrangements are agreed".
Theresa May will be able to persuade her Cabinet to support her preferred Brexit deal - but only after more of her ministers resign, according to a government source.
A Downing Street spokesman said Sunday the Times report was "speculation" and that negotiations with the European Union are ongoing.
"This is all speculation".
"We are making good progress on the future relationship, and 95% of the Withdrawal Agreement has been settled".
Insisting that negotiations are "ongoing" at a technical level, the Commission's chief spokesman summed up progress by telling reporters: "Not there yet".