In a phone conversation with the Taoiseach, Mrs May said that any agreement would have to include a mechanism to bring an end to the backstop - designed to ensure there is no hard border in Ireland if the UK and EU fail to reach a broader trade deal.
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 cannot be ditched by one side acting alone.
The backstop has become the major obstacle to agreement on the UK’s withdrawal from the European Union, due to take place in March next year.
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Downing Street proposals for a review mechanism are understood to be designed to allay Conservative concerns over the Prime Minister’s plan for the whole of the UK to remain temporarily in a customs union with the EU after Brexit.
Former foreign secretary Boris Johnson has denounced the PM’s blueprint as “an absolute stinker”, arguing that the arrangement was a “national humiliation” which would leave the UK as “a vassal state, a colony”.
Meanwhile, the European Commission dismissed reports over the weekend that a Brexit deal has been privately agreed after major concessions from Brussels.
Insisting that negotiations are “ongoing” at a technical level, the Commission’s chief spokesman summed up progress by telling reporters: “Not there yet.”
Expectations are rising that UK negotiator Olly Robbins will be pressing hard to finalise a deal in Brussels this week, to set the scene for a special Brexit summit later in the month to secure the approval of the leaders of the 27 remaining member states.
Mrs May is expected to brief the Cabinet on progress in talks when members gather for their weekly meeting at 10 Downing Street on Tuesday.
But the EU’s chief negotiator Michel Barnier is spending three days away from Brussels on visits to Finland and Slovakia.
A Downing Street spokesman described Mrs May’s conversation with Mr Varadkar as “constructive”, adding: “They agreed that the intention was that the backstop should only be a temporary arrangement and that the best solution to the Northern Ireland border would be found by agreeing a future relationship between the UK and the EU.
“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.”
A spokesman for the Taoiseach said: “The Prime Minister raised the possibility of a review mechanism for the backstop.
“The Taoiseach indicated an openness to consider proposals for a review, provided that it was clear that the outcome of any such review could not involve a unilateral decision to end the backstop.
“He recalled the prior commitments made that the backstop must apply ‘unless and until’ alternative arrangements are agreed.”
The conversation followed a Daily Telegraph report that Brexit Secretary Dominic Raab has privately demanded the right to pull Britain out of the Irish backstop after just three months.