The Prime Minister was forced to postpone a vote on her plans in December after it became clear the deal would be rejected by MPs.
Labour have said they will back the amendment, prompting speculation that ministers will be forced to accept it in order to avoid a damaging defeat.
Meanwhile, one group of cross-party lawmakers is working on a plan to derail any prospect of a no-deal Brexit, which appears to be the most likely option if May's gambit fails.
Writing in the Guardian, Ms Cooper said a no-deal Brexit would cause "deep and long-lasting" damage, and the country "can't afford to play Brexit chicken and wait to see who blinks first".
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".
Speaking at a technology fair in California, Mr Fox said it would not take the possibility of a no-deal exit off the table. And Downing Street said the Government was "satisfied" with the number of HGVs taking part.
The council's highway manager Toby Howe said carrying out the DfT-led trial earlier "wouldn't have really made any difference", adding: "We've still got a couple of months to make any changes we need to make in time".
She later announced the vote would go forward in the week beginning January 14.
Last month, May pulled a vote on the brokered withdrawal agreement, settled on in November after more than a year of back-and-forth negotiations between London and Brussels, acknowledging it would have been roundly rejected by the UK's lower chamber House of Commons.
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If the deal is voted down, Britain risks crashing out of the European Union on March 29 with no agreement in place, a messy outcome that could plunge the country into its worst recession for decades.
The "war game" was criticized by lawmakers as a waste of time and money and mocked on Twitter as "a fake traffic jam.to show the European Union we are ready for no deal".
Fears about economic disruption Monday sparked roughly 200 legislators including some from the prime minister's Conservative Party to write to May asking her to rule out the no-deal scenario.
"I would say don't let the search for the flawless become the enemy of the good because the danger there is that we end up with no Brexit at all". I think the situation - as it always does - has developed, it evolves.
The move would be meant to limit the scale of the parliamentary rebellion against May.
She did not answer whether she would, if defeated, make a second attempt to get the deal passed by parliament.
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.
"But there is no doubt that if the clause were passed, it would represent a huge challenge to the authority of the government and increase further pressure to find a negotiated way out of the current deadlock".