Foster's pro-British unionist party fears that the backstop provision, if implemented, could one day threaten the province's place in the United Kingdom altogether.
She said Sir Patrick's inquiry, which will publish its report next year, had "been hard for many individually and painful for the party collectively".
However, Mrs May did receive some support when Japanese Prime Minister Shinzo Abe welcomed the fact she had reached a deal with Brussels in a telephone call between the two leaders.
In reality, any bilateral agreement between the United Kingdom and the US is likely to take years to negotiate given its complexity, differing standards in areas such as agriculture, and the fact it would require ratification by the US Congress. Today, as leader of the party, I apologise. What we will be saying very clearly to her that she needs to get rid of the backstop.
Mrs May will attend a community event at Queen's University Belfast, for round-table discussions with groups from across society, including students, academics, community and religious leaders.
"That leaves an election, second referendum or no deal as the only options", he said, "No deal would be disastrous for Northern Ireland, but the DUP wouldn't be unduly troubled by any of those options, unless - as is just possible - a no-deal exit plays badly with its own pro-Brexit support base".
European Union leaders have repeatedly said they are not willing to renegotiate the withdrawal agreement, and observers have suggested Britain would have to make major concessions elsewhere for it to be changed.
The party's leader, Arlene Foster, flatly reiterated the DUP's opposition to the deal "as it now stands", telling May to "stop wasting time" and seek a better one, while also warning of the consequences if she did not.
May has said the European Union will not renegotiate and that Britain could face a hugely disruptive "no-deal" Brexit if it does not take the deal on offer.More news: 'I don't believe' government climate report finding
More news: Salah denies Egypt coach exit claims: I will win trophies for Liverpool
More news: Vettel: "F1 will not be poorer without Alonso"
'The reality is that if we are to secure a better outcome than is now on offer, then the only option is to look beyond this current draft withdrawal agreement and work in the time ahead for an improved outcome'.
The rhetoric highlights the domestic challenges that lay ahead for Mrs May.
"We will vote against it if its brought to parliament", Foster told Ireland's RTE television in an interview from the conference, in her most direct comments on the issue to date.
Foster "is doing well with Brexit, she will always have my support", said Bert Johnston, 81, a party member for 40 years.
"In terms of the confidence-and-supply deal, the DUP can certainly pull the plug on the government but most of the non-Brexit legislative agenda is anodyne - the thinnest program I've ever seen - so that wouldn't be as dramatic as might be presumed", said Tonge.
This weekend it is the Democratic Unionist Party conference in Northern Ireland and it is fair to say it is probably attracting more attention than normal.
"We are heading, under this deal, for Brexit in name only, or the break-up of the United Kingdom", he added.
"Our agreement is of course with the Conservative Party".
The party's Westminster leader insisted the union of Great Britain and Northern Ireland is "non-negotiable". Downing Street said yesterday the Prime Minister was, for example, looking to improve the provisions for security co-operation.