Mr Varadkar said he was hoping to see a Brexit deal signed by the end of the year.
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.
Mr Davis said: "If the attorney-general has warned - as has been reported - that the United Kingdom will not know when the backstop would end and compared it to being stuck in Dante's first circle of hell, then it's time to put an end to this nonsense once and for all."Mr Davis said: "We need the cards laid on the table so we can form a judgment".
May is set to discuss the proposals with her Cabinet on Tuesday, and is likely to attempt to win backing for the plan by warning Brexiteer ministers that they will be personally responsible for a "no-deal" Brexit if they fail to support it and telling remainers that, while she fought for the closest possible alignment until the end, negotiations mean Britain will not be a member of the customs union indefinitely.
"The prime minister raised the possibility of a review mechanism for the backstop", it said.
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And EU deputy chief Brexit negotiator Sabine Weyand backed Ireland when it warned that a time-limited backstop on the Irish border would never be acceptable.
The EU has proposed a backstop that would mean Northern Ireland staying in the EU customs union, large parts of the single market and the EU VAT system.
Asked about the report, a spokesman at May's office said: "This is all speculation".
"Trying to appease the right of the Tory Party has failed".
The deal keeps open the possibility of Britain securing a future free trade agreement similar to the one signed between Canada and the European Union in 2016. Appears any old deal now better than no deal'.
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"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". The prime minister has been clear that we are making good progress on the future relationship and 95% of the withdrawal agreement is now settled.
In practice, the backstop will only take effect if trade talks between the United Kingdom and European Union fail to produce a deal deep and comprehensive enough to avoid any checks on the Irish border. "I certainly hope we are".
Reports of a secret Brexit deal come just as more than 70 business leaders signed a letter calling for a second referendum after warning that a bad deal could threaten the United Kingdom jobs industry.
More than 70 business leaders - from lastminute.com founder Martha Lane Fox and former J. Sainsbury PLC chief Justin King to Cobra Beer founder Karan Bilimoria and ex-chairman of Rolls-Royce Holdings PLC Simon Robertson - signed a letter arguing that both the government's current plans for Brexit, and a no-deal Brexit, would be bad for companies and jobs.
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