The Tanaiste says Dublin will not back any Brexit deal that deviates from the core principles of this week's ill-fated draft UK/EU agreement.
Simon Coveney said his administration would consider new proposals from London, but emphasised it would not agree to anything that fell short of the assurances it needs over the shape of the border post Brexit.
Speaking in the Dail he said, "We are in a position where we still need to find a way forward but, let me be very clear, the core issues that Ireland got agreement on at the start of this week are not changing."
The Prime Minister is under pressure to make progress in talks with Brussels amid an impasse over the border.
Mrs May is expected to present a new text on the border deal, which she hopes will satisfy both the government of the Republic of Ireland and the DUP, who effectively vetoed the deal earlier in the week.
Mr Coveney insisted Ireland wanted a solution that would benefit people on both sides of the border, as well as in Great Britain.
"There are lots of issues that are subject to negotiation," he said.
"We accept that the British government is trying to move this process forward in good faith.
"We want to work with them on that, and not against them.
"But Ireland has real concerns and they are important to this country and its future and they are important to this island and its future and we have an obligation to ensure we act accordingly," he said.
The senior Fine Gael TD added, "We will not support anything that in our view risks the re-emergence of a hard border on this island in the context of these negotiations."
Sinn Féin has urged Mr Coveney to go further, and state that his government would not sign up to any deal that saw Northern Ireland leave the formal customs union and single market structures.
The Tanaiste declined, adding, "This is a sensitive negotiation at a very, very sensitive time so what you are not going to get from me is statements that are going to stoke up what is already a difficult relationship management exercise."
After talks in Dublin on Wednesday night, Irish Taoiseach Leo Varadkar and Dutch PM Mark Rutte made clear the EU would not compromise and allow the Irish border to be kicked down the road to phase two of the talks, even under threat of Britain crashing out with no deal or divorce negotiations dragging on to 2018.
And after a phone call with Mrs May on Wednesday, Mr Varadkar said the PM was hoping to return with a new formal written offer by Thursday, but warned, if there was no agreement, talks would be picked up in the New Year.