The backstop, an emergency fix aimed at avoiding controls on the sensitive border between the British province of Northern Ireland and European Union member Ireland, was the most contentious part of the deal Mrs May agreed to in November.
Sky News reports that not all of the DUP MPs are "on the same page" in regard to how the party will vote on Tuesday evening's meaningful vote.
Brexit-supporting MPs had said they would look at what Mrs May achieved before the vote, but that she would have to show a clear way for the backstop to end.
However, he said the situation was unchanged if a deal that supersedes the backstop can not be reached "through no such demonstrable failure of either party".
She told parliament that she has struggled with the need to honor the 2016 Brexit referendum results while also getting a good deal from Brussels, adding that if MPs vote to leave with no deal, that will become official government policy.
May is hopeful that she can push her deal through in the parliament.
Ranko Berich, head of market analysis at Monex Europe, said: "we are now back to a base case where May's deal is rebuffed and she is forced to seek an extension".
That means there's a big risk the European Union would reject a request for such a brief extension, unless the United Kingdom could clearly explain how it would use the time constructively. However, if MPs vote no to a longer extension then it could end with a no-deal Brexit or deadlock.
Pro-Brexit politicians in May's Tory party insist that the plan - known as the backstop - threatens to trap the United Kingdom inside the EU's trade regime forever, because it would be impossible for Britain to leave. Afterward, hard-core Brexit supporters in May's Conservative Party and the prime minister's allies in Northern Ireland's Democratic Unionist Party both said they could not support the deal.
He described the texts as a "legally binding instrument on the Withdrawal Agreement and protocol on Northern Ireland and a joint statement to supplement the political declaration".
Westminster is waiting to see whether Cox agrees that "legally binding" changes to the agreement secured last night are enough to ensure the Irish backstop will not be be permanent. Two years after Brits voted for Brexit and May set the withdrawal in motion, the U.K.is still squabbling internally about what it actually wants-there's no evidence that a few more months will make a difference.
After the vote, Labour leader Jeremy Corbyn accused May of running down the clock on Brexit and said he will try to get MPs to back a Labour plan. "It's time that we have a general election and the people can choose who their government should be".