Critics said the amendments to the Customs Bill laid down by Eurosceptics on Monday would undermine the UK's recently-announced negotiating position.
No customs border for Northern Ireland.
The newspaper reported that Conservative whips, who enforce discipline in the party, issued the warning to pro-EU MPs, led by the former ministers Stephen Hammond and Nicky Morgan, minutes before a crucial vote on customs last night.
But 12 Conservatives broke ranks to back the customs union measure, even though it is understood that Tory whips told would-be rebels that there would be a confidence vote that evening if it passed.
"It essentially means that, for now, Theresa May will not be facing a no-confidence vote", she added.
The British government has survived an attempt by pro-European Union conservative lawmakers to change its post-Brexit trade strategy.
Remainers led by Stephen Hammond and Nicky Morgan will be pushing a new amendment to the government's trade bill, which will make it a negotiating objective to "participate after exit day in a customs union with the EU", in the event that a deal on frictionless trade can not be secured by January.
"This white paper is a weak-kneed response to European Union because the prime minister is trying to fit it around what she thinks the European Union will accept".
The amendment from Conservative MP Philip Lee, who resigned from the government in protest over Brexit, requires the government to seek continued United Kingdom participation in the EU medicines regulations involving the European Medicines Agency.
Mrs May told MPs the Brexiteer amendments she accepted did not change her Chequers blueprint for Brexit.
After reprieve, Britain's May faces Brexit battle in parliament on...
The European Union must show "flexibility" in the Brexit talks to avoid a "catastrophic" no-deal scenario, the first minister has said.
MPs will carry on debating Brexit on Tuesday when the Trade Bill comes to the Commons.
"Each day that ministers waste arguing amongst themselves increases the risk of the United Kingdom crashing out of Europe without an agreement".
"Not only does this safeguard Northern Ireland's position in the United Kingdom but equally important, it ensures Northern Ireland producers will have free access to our biggest market in Great Britain, thus safeguarding tens of thousands of Northern Ireland jobs".
It was bolstered by the support of four pro-Brexit opposition Labour Party MPs.
The DUP is due to meet the government later this week to discuss the Brexit plan.
Mr Wilson said he hoped the vote would sent a message to Irish prime minister Leo Varadkar and EU Chief Brexit Negotiator Michel Barnier and that it would negate the backstop which he described as a "surrender to the Irish Republic and the EU".
A department spokesman said: "The UK has also repeatedly committed to avoiding a hard border on the island of Ireland, most recently in last week's white paper".
"We would look to be an active participant and this would involve making an appropriate financial contribution".More news: Tour de France: Chris Froome offers Geraint Thomas yellow jersey nod