The row comes as Brexit is expected to dominate the agenda of the British Irish Council, which will be attended by the taoiseach, Leo Varadkar, David Lidington, who is in effect May's deputy, and the Northern Ireland secretary, Karen Bradley.
Minister for Justice Charlie Flanagan said the issue of British Prime Minister Theresa May writing to the DUP was an internal United Kingdom matter, but said there was a real deep feeling of uncertainty among communities with regard to the border.
This would be a huge blow for the PM's hopes of reaching a deal with Brussels by the end of the month.
The response of the DUP has caused frustration in Downing Street, with sources insisting that Mrs May was not hiding behind "weasel words" and had stressed that she would not accept a deal which saw Northern Ireland hived off.
DUP leader Arlene Foster (right) with DUP MP Nigel Dodds.
In the letter, Mrs May explains the EU wants a contingency of Northern Ireland staying in a customs union if no border solution is found during the post-Brexit transition period.
Sammy Wilson, the DUP's Brexit spokesperson, accused May of "total betrayal" and ditching the "promises she made" to the party, which has already threatened to withdraw its support for the embattled prime minister.
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".
"I don't think it leaves it on shaky ground because of course the confidence and supply agreement was entered into at a time of great national instability, we wanted to see stability in the Government at that time and we also wanted to deliver on a Brexit vote that had been taken".
The prime minister has been pushing to agree a UK-wide backstop arrangement, which would see the whole United Kingdom remain in an effective customs union with the EU, after rejecting Brussels' proposal of a Northern Ireland-only solution.
A spokesman added that "the government will not agree anything that brings about a hard border on the island of Ireland".
"And we'll do our best to work through it and make sure we get the best outcome for our citizens".
Speaking to RTÉ news, Foster questioned the motivation for this leak and said her problem with the letter was that May had confirmed a Northern Ireland specific backstop with Chequers for the rest of the UK.
"The only thing I would say is that it is very important that we listen to the voice of Northern Ireland in all of this".
A potential sticking point could be demands for European Union fishing fleets to be given continued access to British coastal waters as the price for agreeing to Mrs May's UK-wide backstop, the Daily Telegraph reported.