DUP leader Arlene Foster has said it came as "a big shock" to the party when they saw the text of the deal the British government was set to agree with the European Union yesterday.
In an interview with RTÉ News, Ms Foster said her party only saw the text of the deal late yesterday morning, despite asking to see if for five weeks.
Ms Foster said that "once we saw the text, we knew it was not going to be acceptable" because the DUP could not sign up to anything that would allow a border to develop in the Irish Sea.
She said that the DUP's red line was a situation where Northern Ireland was different from the rest of the UK, and this had been made "quite clear" throughout negotiations with Mrs May.
Ms Foster also said that she has told British Prime Minister Theresa May that they would not support Brexit legislation in the House of Commons unless the text presented yesterday was changed.
An Irish Government spokesperson has pointed out that the Taoiseach has said on numerous occasions that the deal reached on Brexit yesterday was the agreement.
However, he added there is the possibility of more language being inserted into the Brexit document involving the union.
He said this had not been sought, or offered.
Ms Foster said that the British negotiating team indicated to her that it was the Irish Government which prevented the DUP from being shown a copy of the text.
However, the Government has rejected Ms Foster's claim and said it had "no role whatsoever in the negotiations conducted by the British government.
A Government statement adds that they "had no involvement in any decision on which documents should go to the DUP."
Arlene Foster tells RTÉ News the DUP had requested to see text of Brexit deal for five weeks, only saw it late yesterday morning pic.twitter.com/OxXqOQpQLL— RTÉ News (@rtenews) December 5, 2017
Ms Foster said that the DUP now wish to look at the text, make it clear what they cannot agree with and try to work through to move on to Phase 2 of the talks.
She said that she had a very open conversation with Mrs May after she had made the DUP position clear in a press conference yesterday afternoon.
She said she told Mrs May that "it could have been dealt with differently."
"Nobody wants a hard border on the island of Ireland," she added.
"What we want to see is something that works for everybody, we just feel very strongly that the text that was presented to us would have boxed us into a corner before we’d even gone to the trade negotiations," she added.
Ms Foster added that the Irish Government have looked for a lot of detail in relation to the border.
"They don't need to have that detail to move on to Phase 2 so we can talk about trade," she said.
"They have listened to the UK government, and indeed ourselves, around the fact that we don't want a hard border, but instead of accepting that as bona fide they have decided they need a lot more detail.
"They are pushing at an agenda that leads a lot of unionists to think that there is something else behind all this."
When asked to expand on this, Ms Foster said "it is very clear that there has been quite an aggressive agenda coming from Dublin recently."
Arlene Foster says the Irish Government is 'pushing at an agenda that leads a lot of unionists to think that there is something else behind all this' pic.twitter.com/hMCWwOq2FQ— RTÉ News (@rtenews) December 5, 2017
She said that a lot of people "noted" the moment when Minister for Foreign Affairs Simon Coveney told an Oireachtas Committee he would like to see a united Ireland within his political lifetime.
Ms Foster also said: "Nobody wants to see a hard border, but the reality is there is a border, it's there because we are two different jurisdictions and I think some people need to be reminded of that sometimes."
Meanwhile, Mr Coveney has said that the core meaning of the text must remain, as the Irish Government will not be reversing out of an agreement which they felt they had secured yesterday.
Taoiseach Leo Varadkar has told the Dáil that Ireland cannot move on to Phase 2 of the negotiations without the assurances it has been promised by Britain in relation to the border.
Sinn Féin leader says DUP do not speak for North
Michelle O'Neill, Sinn Féin's leader in Northern Ireland, has told Mrs May that the DUP do not speak for the majority of people in the north.
Ms O'Neill spoke to the British Prime Minister by phone earlier today.
She said that the "democratically expressed will of the people of the North is to remain in the EU."
Ms O'Neill added: "I told Theresa May that the Good Friday Agreement was the clearest evidence of the unique and special circumstances of the island of Ireland."