Aides, the Telegraph reports, are believed to be drawing up a plan to make MPs’ approval of the deal conditional on the European Union providing further concessions.
The move is intended to help limit the scale of opposition to the vote while buying time as negotiations continue with European Union leaders.
A Whitehall source said that while the tone of dialogue between Downing Street and Brussels had improved, they remained “far apart” on what each were prepared to accept.
Senior EU figures are said to be waiting until the last moment before the deadline for reaching a Brexit deal to offer any additional measures, so MPs are left with no time to demand more.
May’s advisers are understood to be considering an amendment making approval of the deal subject to the Government obtaining assurances that the Irish backstop, the fallback plan intended to prevent a hard border, will be temporary.
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 EU there is a majority if they can move a little further.”
She added that such an amendment could allow the government to “return for a third go at the meaningful vote, with a proven recipe for a majority”.
However, as MPs prepare to return to Westminster with a crunch vote looming on an 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.
Addressing opponents on both the Remain and Brexiteer wings of parliament, she said: “There are some in parliament who, despite voting in favour of holding the referendum, voting in favour of triggering Article 50 and standing on manifestos committed to delivering Brexit, now want to stop us leaving by holding another referendum.
“Others across the House of Commons are so focused on their particular vision of Brexit that they risk making a perfect ideal the enemy of a good deal.
“Both groups are motivated by what they think is best for the country, but both must realise the risks they are running with our democracy and the livelihoods of our constituents.”
The prime minister said that the British “genius for pragmatism” had always found a way forward which commands consensus at “moments of profound challenge” such as this.
“The only way to both honour the result of the referendum and protect jobs and security is by backing the deal that is on the table.”
But a former Thatcher cabinet minister, John Redwood, said a no-deal Brexit “will work just fine” despite the “idiotic” warnings about potential shortages of food and medicines.
On the other side of the Tory divide, pro-EU veteran Ken Clarke said May’s deal – which he would be prepared to support – is “dying”, and he would be “amazed” if the mood of MPs had changed over the Christmas break.
Instead, he called for Brexit to be postponed until a way forward can be found.
MPs will resume debate on the Brexit deal on Wednesday ahead of a vote the following week.