This would avoid the EU's "backstop" solution that would have treated Northern Ireland differently from the rest of Britain.
"In order to ensure that the backstop, if ever needed, would be temporary, the prime minister said that there would need to be a mechanism through which the backstop could be brought to an end", the spokesman added.
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.
The plan also means that May is now in a position to issue an ultimatum to Brexiteer rebels within her party, the Sunday Times said.
Downing Street has called the reports speculation, but also claim that the majority of the UK's Brexit plan had already been agreed.
Meanwhile, the Daily Telegraph reported that Brexit Secretary Dominic Raab has privately demanded the right to pull Britain out of the EU's proposed Irish backstop after just three months.
Simon Coveney, the Irish foreign minister, made clear on Monday morning that this would never be agreed.More news: Amazon drops free-shipping minimum for all customers
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"This has been committed to by the United Kingdom in order to have a Withdrawal Agreement".
Waterstones chief executive James Daunt, ex-Sainsbury's chief executive Justin King, Lastminute.com founder Baroness Lane-Fox and Innocent Drinks co-founder Richard Reed were among signatories of a letter calling for a People's Vote on leaving the European Union.
In a major intervention on the controversial backstop, amid reports that the government plans to keep Northern Ireland in aspects of the European Union trade structures, Mr Davis said it was "pretty clear there is genuine and significant concern regarding the implications of any fresh backstop text". But that goal has been complicated by May's intention to take Britain out of the EU customs union and single market.
"My guess is that the customs union style deal that Theresa May is lining up. she'll get it through parliament", he said.
"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".
Mrs May has flatly rejected the idea, saying she would not agree to anything that risked splitting the UK.
Even if a deal is done in Brussels in the coming days, May will have to sell it in London - first to her own Cabinet, and then to Parliament.
The online poll of 20,000 people, conducted by Survation, estimated Remain would win another in/out referendum by 54%-46% and that 105 local authority areas that voted Leave in 2016 would now be carried by the Remain side.