"In the face of the EU's unwillingness so far to move, it is the duty of the U.K. Government to stand firm" against the EU's "bad deal" Democratic Unionist Party deputy leader Nigel Dodds said in a statement.
"If the deal is not voted on [in parliament], then we are going to be in uncharted territory", she said.
Amid doubts over whether the deal will have enough backing, Prime Minister Theresa May used her New Year's message to rally support.
"Others across the House of Commons are so focused on their particular vision of Brexit that they risk making a flawless ideal the enemy of a good deal".
The vote has been rescheduled for the week starting January 14 after a debate which is scheduled to begin in the coming week.
Nikki da Costa, who was Downing Street's director of legislative affairs until November, said: "Getting conditional approval isn't enough for the Government to go ahead but it may be enough to show the European Union there is a majority if they can move a little further".
'At that stage it makes sense to go to the country and say here we are, this is what we've managed to negotiate, this is the deal that we have managed to conclude because we don't have the same red lines as Theresa May, we think it's a better way forward.
"If some of those MPs can be persuaded to back the Prime Minister's deal, it won't be because they've come under pressure to do so from their local party members over the Christmas break".More news: Trump turns to 'Game of Thrones' to push for border wall
More news: Carragher suggests formation change after City defeat
More news: India have dominated Australia ever since the start of Border-Gavaskar Trophy
She said a second referendum would be divisive and disrespectful to those who voted to leave in the initial vote, and also highlighted the lack of time available to hold a new referendum.
Barry Gardiner, the shadow worldwide trade secretary, said Labour could campaign on a promise to negotiate a better Brexit deal than that secured by Theresa May, and said he personally believed that any such deal could then be put to the public.
Another former minister said: "My wife said "I really can't be arsed, I've been to Downing Street before" so I will be going along alone just to see what approach they are taking".
As MPs prepare to return to Westminster with the crunch Commons vote looming on the Withdrawal Agreement thrashed out with Brussels, the Prime Minister said no alternative plan was able to respect the 2016 referendum result, protect jobs and provide certainty to citizens and businesses.
The Daily Mail reported the PM is working on a "double lock" to put a time limit on the backstop.
"In fact we're more alarmed about what is coming out from the European Union and especially the Irish government", the Democratic Unionist Party's Sammy Wilson said when asked if he was reassured by signals from Brussels.
Labour's policy is to push for a general election if the Prime Minister fails to get her Brexit deal through Parliament.
Mr Davies told BBC Radio Wales' Sunday Supplement programme that it is "very unlikely" Mrs May will win the vote.