In a letter to the DUP sent this week, May said the EU was insisting that a backstop which keeps Northern Ireland wedded to the single market and customs union is included in the deal now being negotiated in Brussels.
The Northern Irish party, which May relies upon for her Commons majority, had reacted angrily to suggestions in a letter from the prime minister, which had been leaked to the Times, that Northern Ireland could have a different regulatory regime to the rest of the United Kingdom if the Irish backstop comes into force.
Mr Varadkar's comments came after a leaked letter from Theresa May to the Democratic Unionist Party suggested a border down the Irish Sea could be included in the withdrawal agreement.
According to The Times, the DUP have taken Mrs May's words to mean a Northern Ireland-only backstop arrangement will still be included in the legal text of the UK's divorce deal with the European Union, despite her assurance she will not allow it to "come into force".
The DUP has interpreted the wording of her letter to mean that Northern Ireland-only measures will be contained in the Brexit divorce deal despite Mrs May's insistence it will never come into effect.
The DUP hates the Northern Ireland-only backstop as it would create an array of new checks between Northern Ireland and the rest of the UK.
Mrs Leadsom also insisted there will not be a second referendum when pressed on what would happen if Parliament rejected a Brexit deal, although she stopped short of saying no deal would automatically follow.
Any version of the backstop would apply unless and until a wider UK-EU deal on the future relationship solved the issue of how to avoid a hard border with the Republic.
DUP leader Arlene Foster said May's letter "raises alarm bells for those who value the integrity of our precious Union".
She said: "It cannot be a decision that can be overturned by the European Union, it must be capable for the United Kingdom to decide to leave that customs arrangement and it cannot be something the European Union can then hold us to".
May depends on the 10 DUP MP votes for a majority in Westminster and will likely need them for any vote on a deal she strikes with Brussels.
Britain's culture minister Jeremy Wright arrives In Downing Street.
With less than five months to go until Britain is due to leave the bloc on March 29, both sides remain at odds over how to avoid a hard border in Ireland and are yet to agree a backup plan for the Irish border should a no-deal Brexit occur.
Taoiseach Leo Varadkar, Mrs May's effective deputy prime minister David Lidington and Secretary of State Karen Bradley will attend a summit on the Isle of Man on Friday.
"We are still in negotiations, and on that basis we don't know when and if this will conclude", a spokesman said, adding that a much-rumoured Cabinet meeting to allow ministers to sign off the deal had not been scheduled.