The source said ministers were "looking at all options to secure the vote".
Ministers fear a no-deal Brexit will result in a huge build-up of queues at ports.
"I will certainly not be voting to hand over £39bn to the European Union with nothing in return in a deal which keeps us shackled to the European Union in perpetuity".
Despite voicing support for an alternative, Ms Rudd is still encouraging her colleagues to back Mrs May's deal in Tuesday's vote.
He predicted that no consensus would be found in the Commons and the issue would have to be put to a second referendum.
"For these reasons I will not be supporting Mrs May's deal, and I believe if it is defeated, that Parliament need to consider other options, including potentially a General Election, further renegotiation either by this or another Government, and the option for a possible referendum on the deal".
But Mrs May rejected the claim, insisting the document contained the same information as a shortened statement made to MPs by Mr Cox earlier this week.
He said the campaign's "core message" would be that the PM's Brexit deal is a "total betrayal of trust in democracy".
In an historic vote, the government suffered a humiliating defeat yesterday after a majority of MPs voted to pass a motion declaring government ministers, including Cox, to have been in contempt of parliament.
"Everybody knew where we stood, WTO rules, get on with it".
"They know the history of the Irish starvation and Britain's role in it and the very fact that they would seek to inject that into the Brexit debate is crass and offensive".
"(It) would mean an immediate and probably indefinite loss of some security capability which, despite our best efforts, would likely cause some operational disruption when we leave", he said.
"It leaves the United Kingdom in an indefinite limbo position, with having to continue paying huge sums of money to the EU with no longer being able to influence its decisions, cut off from the economic strength of a common customs union".
Looming over those talks would be the backstop, which could leave the U.K.in a customs union with the EU indefinitely if negotiators fail to agree on another way to prevent border checks on goods and people passing between Northern Ireland and the Republic of Ireland.
"I don't think that's right".
"Are there things which I am uneasy about?"
But MPs voted 311 to 293 to find May's government in contempt on Tuesday afternoon, forcing Downing Street to publish the advice in full.
"The question is do we go into the backstop, do we extend... the transition period?" "What they are voting on is what was agreed in the Withdrawal Agreement and the Political Declaration".
However, Mr Ellis's work is cut out in Northampton it seems, though he says voting against it could well result in Britain leaving the European Union in March with no deal whatsoever.
The advice, from Attorney General Geoffrey Cox, the country's top law officer, examines the legal effect of the proposed backstop arrangement, or Northern Ireland protocol, which is a fallback measure created to maintain an invisible border in Ireland.
"The backstop is not there in order to be used, necessarily". He reiterated Thursday that the May agreement on Brexit was the only one on the table.
Also invited for briefing were the first ministers of Scotland and Wales and representatives of the Northern Irish parties.
The day before the vote, on December 10, the European Court of Justice of Justice will deliver a judgment on whether Britain can unilaterally reverse its move to leave.
MPs have just days before they have to make the decision on whether they want to approve or block Theresa May'sBrexit deal.