The concept of the backstop - an agreement governing the customs status of the Ireland/Northern Ireland border in the event that Britain and the European Union can not agree a long-term relationship by the end of 2021 - has been the main sticking point preventing Mrs May's Withdrawal Agreement from being passed by Parliament.
UK Brexit minister Martin Callanan ruled out that prospect and said May would update MPs on Wednesday about the assurances over the backstop she is seeking from the EU.
The move by Corbyn - himself criticised for his failure to push for a second referendum - came after the First Minister of Scotland and leader of the Scottish National Party (SNP) Nicola Sturgeon said that she would announce her preferred timetable for a second referendum on Scottish independence "when we get to the end of this phase of the Brexit process".
"The closer we get to March 29 without a deal, the more assets will be transferred and headcount hired locally or relocated", he told City AM.
Sources in Brussels have told AFP for several weeks that Britain has been discussing the possibility with European officials, while this was also reported in the Daily Telegraph newspaper on Tuesday.
Kent Police said it escorted the lorries between the key port of Dover in Kent and nearby Manston Airfield, which could serve as an emergency lorry park if any post-Brexit border checks cause major delays for freight vehicles moving across the English Channel to France.
The Government will begin broadcasting radio adverts on Tuesday to help people prepare for Brexit, focusing on topics including passports, visas, healthcare and driving in the EU.
After previous plans for a vote in December were cancelled the day before it was due to happen, Mr Corbyn asked Brexit Secretary Steve Barclay for a guarantee that "faced with yet another humiliating defeat that the Prime Minister won't just run away".
The letter was organised by two West Midlands MPs, the Tory Dame Caroline Spelman and Labour's Jack Dromey.
The "war game" was criticised 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".
"The real choice facing Parliament and the country is now clear", he said.
Some Brexit supporters say a no-deal exit is the only way to truly leave the bloc and that warnings of the economic consequences have been overblown to drum up support for Mrs May's plan. We all want to have a fruitful profitable relationship with the United Kingdom in the future so the backstop is just a last resort solution.
However, it could galvanize support for future amendments on similar bills and prove that there is a parliamentary majority against a no-deal Brexit.
Their goal is to ensure that a "no deal" Brexit could only be delivered with the explicit consent of Parliament - something that is unlikely, given that a majority of lawmakers oppose such an outcome. Eleven Tories had signed the amendment, including former cabinet minister Nicky Morgan, select committee chair Sarah Wollaston, former minister Nick Boles and Sir Nicholas Soames.
As Sky's political editor Faisal Islam tweeted, "this is the start of constitutional trench warfare where Parliament will try to assert its power over the executive to prevent No Deal - highly significant".