Earlier, Mrs May rejected claims by Mrs Foster that she has "given up" on getting a better Brexit deal.
DUP leader Arlene Foster called for a Brexit "third way", insisting the choice is not between the current draft agreement and no deal.
Addressing her party conference in Belfast, Foster reiterated her rejection of the draft agreement.
"I think we have to take a look at, seriously, whether or not the U.K.is allowed to trade, because you know right now, if you look at the deal, they may not be able to trade with us", he said.
Deputy leader of the DUP, Nigel Dodds, also raised threats to Mrs May's plan if Northern Ireland is to be treated as separate from the of the United Kingdom, saying the current agreement could see the United Kingdom "locked into an European Union straitjacket, divided and diminished".
"The DUP is leading the way and the Conservative Party need us - if they don't know this by now they are living in cloud cuckoo land", said Alan Ewart, 68, a member of the party since its inception in 1971 and now its finance director.
"The Titanic springs to mind, and now is the time to point out the iceberg ahead".
Speaking at the Royal Welsh Winter Fair in atrocious conditions, Theresa May said a failure to secure the Brexit deal would lead to more uncertainty and division.
After enduring criticism of the Brexit withdrawal agreement in the Commons on Monday, the prime minister began Tuesday by taking issue with US President Donald Trump's suggestions that the deal could threaten any future US-UK trade deal.
Mrs May said she spoke to Mr Sanchez on Wednesday night and is "confident on Sunday that we'll be able to agree a deal that delivers for the whole United Kingdom family, including Gibraltar".
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The US president also suggested the terms of the settlement would hamper the UK's ability to sign a post-Brexit trade deal with America. "I don't see any circumstances at present where that vote will be a vote to go ahead in Theresa May's favour".
Foster declined to say whether her party would remove its confidence and supply arrangement keeping May in power if the Brexit deal was passed by the Commons, saying only that she would "review" the issue in this case.
Asked if that meant the European Union would wait for the House of Commons to vote a second time on the deal if it doesn't pass in December, Mr Varadkar said he doesn't have "a crystal ball".
The prime minister, who faces an uphill battle to get the controversial deal through Parliament, has been accused by the Orange Institution of "betraying the unionist community" in Northern Ireland.
Deputy leader Nigel Dodds was clear in his speech to the conference on Saturday. And without the support of the 10 DUP MPs, May could lose through votes from opposition MPs only.
The party is threatening to end its support over the Brexit plan favored by May.
May does not now have enough support in the British parliament to ratify the deal.
Foreign Minister Augusto Santos Silva says that the original Brexit guidelines laid out past year that included assurances to Spain that it could deal with London directly on the issue of Gibraltar were "wise".
"But right now, as the deal stands, they may not be able to trade with the US, and I don't think they want that at all". "She has maintained our red line over the integrity of our U.K".