The party’s deputy leader Nigel Dodds made clear the DUP was ready to continue voting against the Government.
He called for the Prime Minister to ditch her draft EU divorce proposals and “work for a better deal”.
It comes after the party yesterday delivered warning shots over Brexit, accusing Mrs May of breaking fundamental promises.
It appeared to call into question the future of the "confidence and supply" arrangement by which the DUP props up Mrs May's minority Conservative Government.
Under the terms of the deal, agreed after Mrs May lost her Commons majority in last year's general election, the Northern Ireland party is supposed to back the Government on Budget matters and on confidence votes.
The move piles further pressure on Mrs May as she prepares to meet EU leaders for a final crunch meeting in Brussels on Wednesday.
The Prime Minister will hold talks with EU President Jean-Claude Juncker as she scrambles to close her Brexit divorce deal.
Mr Dodds said that the agreement with Conservatives committed the DUP to pursue the shared objectives of strengthening the Union and seeking a Brexit that benefits all parts of the UK.
While the DUP had "kept to our word" on the agreement, the resignation of several Conservative ministers showed that Mrs May's deal "does not represent those shared objectives", he said.
He added: "The Government will require DUP support to deliver its domestic agenda. We will continue to use our influence for the good of everyone across the United Kingdom.
"If the Government can look beyond a Withdrawal Agreement, which is uniting people from across the political spectrum against it, and instead work towards a better deal, then an outcome can be delivered that truly works to benefit all parts of the United Kingdom."
Jacob Rees-Mogg warned earlier that the Conservatives will be stuck with Mrs May as leader for the next general election unless they move to get rid of her now.
Amid signs the attempt by Brexiteers to force a vote of confidence in the Prime Minister has stalled, Mr Rees-Mogg acknowledged they were struggling to get the support they needed.
However, the leader of the pro-Brexit European Research Group (ERG) insisted there was little enthusiasm among Tory MPs for Mrs May to take them into the next election, due in 2022.
"I think it is now or the Prime Minister will lead the Conservatives into the next election," he told reporters at a Westminster news conference.
"You find MPs privately who will say to you they think that is a really good idea in any number and I would be quite surprised."
Mrs May’s meeting in Brussels comes ahead of a special EU Brexit summit on Sunday.
In a move likely to be welcomed by Brexiters, Number 10 confirmed that the Government would look at potential technological solutions to keep the Irish border open when Mrs May meets leaders for talks.
Mrs May's official spokesman confirmed that references in last week's draft agreement to "alternative arrangements" for the border could involve technical measures of the kind previously promoted as the "maximum facilitation" solution, or Max Fac.
Last week, senior ERG figures were confidently predicting they would get the 48 letters of no confidence from MPs needed to trigger a vote in Mrs May's leadership
But with Brexiteers apparently divided over whether it was the right time to mount a challenge, the prospect of them achieving their target appeared to be dwindling.
However, Mr Rees-Mogg warned that they needed to consider whether they really wanted to carry on with her at the helm.
On Monday night, the DUP joined Labour in voting against the Government on a Budget measure while abstaining in two other divisions in apparent breach of their "confidence and supply" agreement.