Sir Bernard said: "These amendments were only ever about clarifying Government policy, that the United Kingdom will not join a customs union with the EU, nor agree to split Northern Ireland from the United Kingdom for customs purposes, that we are leaving the EU VAT regime, and any customs arrangement with the EU is not one-sided".
"The only solution is to take the final Brexit decision out of the hands of deadlocked politicians, away from the backroom deals, and give it back to the people", she wrote in an article in The Times.
Although the measures are unlikely to pass in the absence of Labour support, it could prove an opportunity for a show of strength by the rebels meant to pressurise her into retreat.
Her recent white paper outlining plans for a common rule book with the European Union over trade in goods has infuriated those who favour a complete break even if it risks causing an economic shock.
The size of the threat to her position should become clear on Monday when euroskeptic MPs put forward a series of proposals to toughen up the government's customs legislation during a parliamentary debate.
A Labour source told HuffPost: "Theresa May sounds like she is running scared of her own MPs".
Some MPs are planning to vote for amendments that would tie Britain to a permanent customs union with the EU.
Former education secretary Justine Greening said other senior Conservatives quietly back a second Brexit referendum.
She told MPs: "He is absolutely wrong in his reference to the agreement that was reached at Chequers, I would not have gone through all the work that I did to ensure that we reached that agreement only to see it changed in some way through these Bills".
In one of the rowdiest debates in parliament on Brexit so far, pro-EU Conservative lawmaker Anna Soubry said: "The only reason that the government has accepted these amendments is because it is frightened of somewhere in the region of 40 members of parliament, the hard no-deal Brexiteers, who should have been seen off a long time ago".
She said others were planning to try and bring down a bill that was essential to enable Britain to prepare for life outside the European Union, which would "risk our ability to make the necessary preparations for a no deal".
A Welsh Conservative MP has quit as a minister so he could vote against the government on its Brexit Customs Bill.
The party's European Research Group says it will reject any last attempts at compromise by Number 10 as they hope to force May to change course over Brexit or risk a no-confidence vote before the summer break by demonstrating the depth of their support.
"I have come to the conclusion that it does not respect the referendum result - it is not what people voted for", he said.
The official was due to be quizzed over his role in drawing up Mrs May's Brexit plan and is said to have clashed repeatedly with former EU Exit Secretary David Davis.