When commenting on the prime minister's letter, he said that it showed that the United Kingdom was not delivering on its promises to Northern Ireland and that a range of concerns had emerged.
Without DUP support for a deal, May will have much less chance of getting her deal through parliament.
Earlier today, Scotland's First Minister accused the UK Government of trying to promise "all things to all people" on Brexit.
But David Lidington, Mrs May's de facto deputy prime minister, said he hoped that once a deal was on the table MPs would rally behind it.
The DUP has interpreted the wording of her letter to mean that specific Northern Ireland measures will be contained in the Brexit divorce deal despite Mrs May's assurances.
"We will not support arrangements that leave Northern Ireland separated from the rest of the United Kingdom and tied to the European Union's customs or regulatory regimes", Foster said in a letter to May which was published on November 9.
"The government will not agree anything that brings about a hard border on the island of Ireland".
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"The PM knows the consequences, she now needs to reconsider", DUP lawmaker Sammy Wilson said.
However he was clear that the DUP weren't the only voices that should be listened to in Northern Ireland.
In it, Ms. May reportedly tried to assure her allies that she would never allow a Brexit deal proposal offered by Brussels to "come into force". 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 Ireland.
A UK-wide customs deal would maintain quota-free and tariff-free access to European markets for the British fishing industry and in return the EU wants to keep continued access to United Kingdom waters for its trawlers, the newspaper said.
Speaking at the British-Irish Council summit on the Isle of Man, Mr Varadkar said he was hopeful a Brexit deal could be done by the end of the year but it would not amount to a "clean break" as talks would have to continue.
As we enter the endgame of the Brexit negotiation process, all pro-Remain parties must continue to represent the demand for our rights and political processes to be protected and the onus is firmly on the Dublin Government and the European Union 27 to ensure Theresa May is held accountable to the agreement she made in December.