However, if Brexit does not go ahead, only 35 per cent said they would support a united Ireland outside of the UK while 52 per cent would vote for Northern Ireland to remain part of the UK, with 11 per cent unsure how they would vote and the rest would not take part in the ballot.
According to the Telegraph, May has included Attorney General Geoffrey Cox, the government's chief legal advisor, in her special Brexit Cabinet after ministers said they wouldn't sign off on a deal without his advice.
The Irish border has proved the biggest obstacle to a deal, with both sides vowing not to create a hard border between Ireland and Northern Ireland for fear of destabilising the peace accord that ended decades of deadly sectarian violence.
May told Varadkar that there would need to be a mechanism through which the backstop could be brought to an end, a spokesman from her office said in a statement.
A spokesperson said: "The prime minister has been clear that we are making good progress on the future relationship and 95pc of the Withdrawal Agreement is now settled and negotiations are ongoing". We want to close that gap.
The newspaper also said Mrs May wants to make enough progress this week to persuade the European Union to call a special summit in November to sign off on the final details.
The talks could extend into December, but Theresa May would rather avoid this scenario as it would ensure that there would be no chance of securing the parliamentary vote over or before Christmas.
Fox, a prominent Brexiteer, said there had already been a referendum.
Ireland insists that there must be no border infrastructure, the DUP insists Northern Ireland must not be treated differently from mainland Britain, and Brexiteers say Britain must have the right to do its own trade deals after Brexit.
Mrs May has reportedly dispatched three Cabinet ministers to Dublin and Belfast in an attempt to quell resistance to her latest Brexit plan.
Still, it's not only ministers Mrs May has to worry about.
Under the banner headline "Revealed: Theresa May's secret Brexit deal", The Sunday Times said the European Union accepts that regulatory checks on goods can be carried out in factories and shops rather than at the border.
Mr Banks once again denied Russian involvement in the campaigns and said funding came from his United Kingdom businesses - and he stuck to his line that he's the victim of efforts aimed at "undermining the Brexit result" by a "group of vicious MPs". Not with a deal apparently so close at hand.