Brexit talks have been deadlocked over the so-called backstop - the fallback position that would be activated if a future EU-UK trade deal does not produce a solution that would prevent a hard border between Ireland and Northern Ireland.
The report claims an "all-UK customs deal" will be written into the legally binding withdrawal agreement, which would do away with the need for the controversial "backstop" arrangement agreed by the United Kingdom last December, which would see Northern Ireland remain in full alignment with the EU's single market and customs union rules in the event of a no-deal Brexit.
Mr Varadkar's spokesman said the Irish premier indicated he was ready to consider proposals for a review mechanism, but only if it was clear that the arrangement can not be ditched by one side acting alone.
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.
It comes amid reports of an imminent deal - which Downing Street has described as "speculation".
"He recalled the prior commitments made that the backstop must apply "unless and until" alternative arrangements are agreed".
"I would have thought there will be quite a lot of Labour MPs in the Midlands, in the north, south Wales, sitting in seats where there was a big leave majority and by voting for May's deal they can say to their constituents "I voted for us to leave the European Union" even if the detail means we haven't really".
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Under the EU's proposal, Northern Ireland would remain in the customs union and in the single market for goods.
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.
"Obviously still having this issue in relation to the insurance arrangements for Northern Ireland and Ireland, and that very much remains our focus and attention in getting that deal".
Britain would remain in a temporary customs union with the EU, avoiding a hard Irish border, as part of a new deal being thrashed out between London and Brussels, according to a Sunday Times report.
The Department for Exiting the European Union (DEXEU) said: "We wouldn't comment on private meetings".
Meanwhile, 1400 of the UK's top lawyers have urged Theresa May and MPs to back a second Brexit referendum, saying that "democratic government is not frozen in time".
They suggested that a December deal would mean not only a later parliamentary vote but would require spending on no-deal planning and changes to the roles of hundreds of civil service.