But 34 Tory MPs still voted against it, as did the DUP, and despite frantic last-minute lobbying by Tory whips, who offered Labour MPs £100 million for their constituencies to back the deal, only five Labour MPs voted with the Government.

The Daily Telegraph understands that Gavin Barwell, Mrs May’s chief of staff, has been told by Cabinet ministers that it is time to give the Cabinet a vote on either no-deal or membership of a customs union.

One Cabinet minister said: "Cabinet needs to vote and decide on the Government position. David Cameron did it before the referendum when he asked ministers to put on record their positions.

"There has to be a vote in Cabinet, we have got to put our names to something.

"The Remainers clearly want a customs union as an alternative but there isn't a majority for that in the party - it would destroy the party.

"We would end up with a Ramsay MacDonald-esque Government where we would be completely at odds with our own party. It's time to be bold, we need to embrace no deal."

As well as Sunday’s conference call, Mrs May is expected to hold a Cabinet meeting on Monday morning when ministers will have a second chance to insist she decides on a plan B.

Later the same day MPs will vote on the most popular alternatives to Mrs May’s deal following a series of “indicative votes” last Wednesday. Although no one idea achieved a majority, the idea of a customs union lost by just six votes and a second referendum was rejected by just 27 votes.

Cabinet ministers are expected to demand a free vote on Monday, having been ordered to abstain over the indicative votes this week.

Government sources said Mrs May hopes that if a customs union or a second referendum proves popular with MPs, hard-core Brexiteers will finally decide to back Mrs May’s deal to avoid a worse alternative.

That would enable her to hold a fourth vote on the deal next Thursday or Friday. Downing Street believes it can find a way of persuading John Bercow, the Speaker, to allow another vote on the deal despite his insistence that it has to be “substantially different” from what has been rejected before.