DUP leader Arlene Foster said the text of Monday's proposed Brexit deal on the border came as a "big shock" to the party.
The DUP vetoed a potential agreement on "regulatory alignment" on the island of Ireland amid fears it would effectively create a border on the Irish Sea.
Mrs Foster claimed that despite requests to view the text for five weeks, the DUP was only allowed to see it on Monday morning, hours before the deal was set to be agreed between the British government and the European Union.
"Once we saw the text, we knew it was not going to be acceptable," she told RTÉ.
The former first minister said that the DUP had made its' position "quite clear" throughout negotiations with Mrs May that they would reject any deal which treated Northern Ireland differently from the rest of the UK.
Mrs Foster said that she has told the Prime Minister that it "could have been dealt with differently" and that she will not support Brexit legislation in Westminster unless the deal due to be agreed on Monday was changed.
The DUP leader, whose party signed a 'confidence and supply' agreement to prop up Mrs May's Conservative government in June, claimed that British negotiators had indicated it was the Irish side who had requested that the DUP was not shown the text.
That was dismissed by the Irish government, who said it had "no role whatsoever in the negotiations conducted by the British government" and "no involvement in any decision on which documents should go to the DUP."
Mrs Foster 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."
The DUP's deputy leader, North Belfast MP Nigel Dodds, earlier also said that the party only received draft proposals on Monday morning.
He said: "Despite several briefings over the course of the last few weeks, we only received written text late yesterday morning.
"We understand this was due in part to delays caused by the Irish Government and the EU negotiating team."
"So the DUP does stand strong for the union and we also issue a warning to the Dublin government - that by continuing its aggressive stance they are in danger of delivering for themselves the very outcomes that they said they want to avoid."
Mr Dodds said there could be regulatory alignment in certain areas, but added "not in relation to following the rules of the single market or the customs union for Northern Ireland as a generality."
Ulster Unionist leader Robin Swann said: "I send this friendly warning to the Taoiseach - you are being played by Barnier and Verhofstadt and when the money's right, you'll be dropped like a stone."
