Andrea Leadsom, a leading Eurosceptic Cabinet minister, warned that the EU can not be allowed to stop Britain from leaving the Customs Union after Brexit.
In the letter, obtained by the Times, Mrs May said: "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 in to force".
The DUP leader said "no unionist" could back Theresa May's apparent advocacy of a withdrawal treaty that includes a Northern Ireland specific backstop measure to avoid a hard border on the island of Ireland.
It comes as tensions are mounting between the DUP and the British Prime Minister over the future of Northern Ireland post-Brexit.
However, in the leaked letter, it is revealed Mrs May admitted to the DUP that the European Union is still proposing a Northern Ireland-only backstop arrangement alongside a UK-wide solution.
The British prime minister wants a "backstop" measure which would create a temporary "joint customs territory" with the European Union for the whole of the UK.
Throwing his support behind a second referendum, he added it would be a travesty not to let voters have their say on the final deal.
The prime minister wrote: "They want to maintain a Northern Ireland-only "backstop to the backstop" in case the future negotiations are unsuccessful".
May's letter to the DUP said that she "could not accept there being any circumstances or conditions" for the Northern Ireland-only backstop coming into effect.
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Zack Greinke ($34 million), David Price ($31 million) and Clayton Kershaw ($31 million) all got more. Whether the Nationals will be without Harper remains to be seen.
The DUP has reacted angrily to a letter from Theresa May to DUP leader Arlene Foster and accused May of breaking promises with regards to an Irish Sea border.
Mrs May's proposal for an "independent mechanism" to solve the customs dilemma was agreed to be the best way to avoid a hard border between Northern Ireland and the Republic by the Cabinet.
The Prime Minister relies on the support of the DUP's 10 MPs for her Commons majority, votes which may become crucial as she attempts to get a deal through Parliament.
One option could be for the whole of the United Kingdom to remain temporarily aligned to the EU's customs union, avoiding the need for customs checks at the border until a free trade deal is ready.
His comments came on the same day Austrian newspapers reported that Brexit Secretary Dominic Raab and European Union chief negotiator Michel Barnier could meet over the next few days to seal an agreement.
Writing in the Sunday Times, shadow Brexit secretary Sir Kier Starmer says he will work with other parties to stop a no-deal scenario.
"I'm hopeful that it can be done in the next few weeks, I think it is more likely than not that we will be able to conclude an agreement in the next few weeks before the end of the year", he said.
Speaking in the Isle of Man, where he was attending a meeting of the British-Irish Council, Lidington said he believed a "new dynamic" would emerge, once MPs saw the full text of the proposed agreement.