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.
Other UK government sources indicated that the cabinet was close to agreeing the mechanism that could end a temporary customs arrangement with the EU.
The Leave-supporting lawyer ran the Cabinet through the legal arguments behind the backstop, which is created to avoid the imposition of a "hard border" on the island of Ireland if talks on a future trade deal failed.
Mr Barnier himself said a breakthrough on the Irish border issue was not close.
Following the Cabinet meeting, Mrs May's spokesman said: "The Prime Minister said she was confident of reaching a deal".
Christopher Vecchio, an analyst at Daily FX, said: "UK Brexit minister Raab leaves meeting and gives a "thumbs up" - $GBP gets a bump across the board".
"There is still a real point of divergence on the way of guaranteeing peace in Ireland, that there are no borders in Ireland, while protecting the integrity of the single market", he told a Belgian TV station.
A senior member of the Northern Irish Democratic Unionist Party earlier said it looked like Britain would exit the European Union without a deal, indicating domestic political headwinds togetting a deal through a split parliament with less than five months to go to Brexit day.
Driver Charged With Killing 2 Children In Park Slope Crosswalk Found Dead
The triple-killing saw Bruns indicted by a grand jury on charges of manslaughter, criminally negligent homicide and assault. Her health issues had become such that this past January she went on medical leave from her job at ClearCaptions.
The Withdrawal Agreement is 95% complete, but the final outstanding issues need to be resolved within the week if the United Kingdom government is going to achieve its aim of finalising a deal this month. "The prime minister has been clear that we are making good progress on the future relationship, and 95 percent of the withdrawal agreement is now settled and negotiations are ongoing".
He said that the Irish Government was working hard to try to reach an agreement by the end of the year but added that it can not countenance the idea of a three-month limit to the backstop agreement.
Sterling fell after a senior member of the Northern Irish party which props up May's government said the United Kingdom looked likely to leave the European Union without a deal.
Michel Barnier has warned that a Brexit deal is "not close" and challenge Thereda May to make the tough choices necessary to get it over the line in the coming weeks.
"Can't understand why Irish government seems so intent on this course".
Cautious optimism that a deal may be in the offing has also been dampened by uncertainty over whether it would pass the British parliament, deeply split between eurosceptic and pro-EU lawmakers, even within May's Conservative Party.
More than 70 business leaders, including lastminute.com founder Martha Lane Fox and former J. Sainsbury Plc chief Justin King - 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.
However, once they return to Northern Ireland pet owners will have to go through a similar process as the certificates do not allow pet owners to move freely between north and south of the border.