If the deal is rejected, it is unclear what happens next, with May insisting her deal was best for the country.
The revelations, reported in The Sunday Times, will spark fury among Brexiteers who have warned of a plot by Remainer MPs to thwart Brexit.
Meanwhile, Jeremy Hunt, the current foreign secretary, is also reportedly making a pitch for the top job as the man who can deliver a "managed no-deal" exit from the Brussels bloc if Mrs May's plan fails.
With the PM's plans plunged into turmoil, civil servants have war-gamed two versions of the United Kingdom holding another referendum.
It also said May's deal had avoided making choices about the future relationship with the European Union, meaning that negotiations would be needed which would likely go on for "a number of years". "She is so committed to her deal, and a second referendum could now be the only way of getting it".
Cabinet ministers have told the Observer that attempts to convince May to delay the vote to avoid one of the largest and most humiliating defeats in recent parliamentary history had not been heeded.
The Work and Pensions Secretary became the first Cabinet minister to publicly discuss the merits of a "Plan B" if Mrs May crashes to defeat in Tuesday's crunch Commons vote.
But the pressure on May mounted over the weekend after Conservative MP Will Quince quit his government role on Saturday in opposition to her deal, and the Sunday Times said further resignations were expected.
A No 10 spokesperson said the vote was going ahead on Tuesday as planned and that any other suggestions were "speculation".
He told BBC's Andrew Marr Show: "The real problem with the backstop arrangement is it gives the power to Brussels and to all the other European Union member states effectively to blackmail us and to get what they want out of the future trade negotiation".
She has not backed Mrs May's deal yet but has said she supports the Prime Minister.
"Why hasn't this point been pressed home during negotiations", she told "The Times" in reference to the report. Rudd argued that a Norway-style soft Brexit could be the way through the impasse.
MPs will be able to spell out potential plans and vote on them - giving a strong indication of where opinion lies.
Former Brexit secretary Dominic Raab refused to rule out a leadership bid, relling Sky News he would not "get sucked into that debate". A group of Cabinet ministers have been mulling it over as a fallback plan.
"But it's better than the governments proposal, but it would not be accepted politically in the UK".
However, critics say it has the drawbacks of keeping free movement, - and tightly limiting the possibilities for doing trade deals elsewhere.
Amber Rudd says her preferred Plan B is a Norway style arrangement, which would see the United Kingdom stay in the single market.
The Remain-supporting frontbencher said a so-called Norway-plus model was a "plausible" alternative.
But it would have huge political force, and it might be very hard for ministers to face down the calls.
"It is her policy which has failed and for which she is accountable".
But the report says this would "result in immediate barriers to UK-EU trade in goods and services".