"I was in Dublin a few days ago".
Arlene Foster says it has "raised alarm bells" as Mrs May appears "wedded to the idea of a border down the Irish Sea".
The European Union's fallback proposal aimed at avoiding a hard border between Ireland and the United Kingdom would effectively keep Northern Ireland aligned with Brussels' customs union and single market.
But the DUP has interpreted the wording of her letter to mean that the EU's proposal - known as the "backstop to the backstop" - will be contained in the Brexit divorce deal despite Mrs May's insistence it will never come into effect.
It comes as tensions are mounting between the DUP and the British Prime Minister over the future of Northern Ireland post-Brexit.
"The government will not agree anything that brings about a hard border on the island of Ireland".
"I am clear that I could not accept there being any circumstances or conditions in which that "backstop to the backstop", which would break up the United Kingdom customs territory, could come into force".
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In the leaked letter, the prime minister also told the DUP she "would not accept being kept in a backstop arrangement indefinitely", while it would be "totally unacceptable" for a time limit to the UK-wide backstop arrangement to then simply result in the Northern Ireland-only solution being adopted.
Sammy Wilson, the representative of the party for Brexit, said: "From this letter it is clear that the government has decided on a plan reverse for Northern Ireland".
Mr Varadkar also said: "I've no specific concerns about the communications that are happening between Prime Minister May and the DUP".
One of the scenarios being discussed is Britain remaining in a customs agreement with the European Union for a limited period even after the end of the transition period on December 31, 2020, which would avoid the introduction of new border controls between Ireland and Northern Ireland, Walker said.
"Similarly, I don't think the relationship between Scotland and Ireland has ever been stronger than it is at the moment".
Ms Sturgeon also said that if a deal proposed by the UK Government can not command a majority in the House of Commons, it should not mean that departing the European Union with no deal is "inevitable".
"People will need to ask themselves what is it that is going to be in the best interests of those who sent them to Westminster to represent them, to ensure that we maintain living standards and investment and prosperity and employment in our country".