The PM is being squeezed on all sides with Brexiteer MPs, Cabinet ministers and European leaders putting pressure on her.
She is keen to use an EU summit next week to secure Europe's backing for her Chequers proposals.
If the meeting goes as badly as last month's gathering in Salzburg, backbench MPs will ramp up their efforts to get Mrs May to dump the plan altogether.
Tory Brexiteers warned today that the "Irish backstop" – a proposal to keep Britain in the customs union if there is no other solution to the Irish border problem – must expire by 2022.
Otherwise Britain will be tied to the EU indefinitely and will not really have left, MPs told The Times.
EU officials declared over the weekend that the two sides are 90 per cent of the way to securing a deal.
But Westminster insiders have said Mrs May must endure a "hell week" as she tries to pin down the final ten per cent.
The PM is hoping to sign off on a draft agreement when she meets European leaders in Brussels next Wednesday.
The deal – which would only contain an outline of the future trade agreement – could then be confirmed at a special summit scheduled for next month.
Mrs May will make her case to the Cabinet tomorrow, where some senior ministers are expected to demand that she prepare a Plan B for if Chequers collapses next week.
One source told the Daily Mail: "If there is another Salzburg-style rejection at the summit then things will start to get fruity."
Even if the PM does cut a deal with the EU, she faces a tricky time getting it through the Commons
No 10 is trying to win over Labour MPs, arguing they should support her proposals rather than risk a No Deal scenario.
But even anti-Corbyn Labour figures have rejected the idea of propping up the Tories in power.
And Tory Brexiteers say Mrs May risks a huge backlash if she continues to pursue the tactic.
Stewart Jackson, former chief of staff to David Davis, warned last night: "This is a waste of time.
"These Labour MPs won’t prop up a Conservative Government but the tactic will alienate many more Conservative MPs and members."
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