"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.
Arlene Foster says it has "raised alarm bells" as Mrs May appears "wedded to the idea of a border down the Irish Sea".
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.
Brexit is expected to dominate the agenda of the British Irish Council, which also involves the first ministers of Scotland and Wales, Nicola Sturgeon and Carwyn Jones.
Current EU proposals state that United Kingdom officials would be "competent authorities" to carry out checks, but in line with EU rules, something the DUP has declared a red line in any Brexit deal they agree to.
Arlene Foster said her party would "not be able to support this if it came to Parliament in the form that it is in the letter".
DUP Leader Arlene Foster at Stormont in Belfast, Northern Ireland, on November 2, 2018.
The Taoiseach says a deal on Brexit is still possible within weeks.
DUP Brexit spokesman Sammy Wilson told BBC Radio 4's Today programme "we want to trust the Prime Minister" but "you have to judge any promise by what is actually delivered in an agreement".
But she told the DUP that the European Union is still pushing for a so-called "backstop to the backstop" which would keep the province in regulatory alignment with the Republic of Ireland to avoid a hard border, the Times reported.
"She has sent us where she believes she is now at, and remember this is before she goes to Brussels to negotiate with them on what they believe is possible, but now, as it stands, we could not support her proposals".
"Secondly, it would be a legal agreement which the government of the United Kingdom could not walk away from - that could only be broken if the government in the United Kingdom and the European Union agreed to it being changed".
"The most important thing for me is the objective and that is to give everyone in northern Ireland and Ireland assurance that a hard border will not develop between north and south no matter what else may happen in the years ahead", Varadkar said.
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