Speaking at the G20 in Buenos Aires, European Council president Donald Tusk warned that Mrs May's Withdrawal Agreement is "the only possible one" and voting it down will either lead to a no-deal Brexit or no Brexit at all.
Mrs May secured an agreement with European Union leaders on Sunday that will see Britain leave the bloc on March 29 but with continued close trade ties.
The East Surrey MP, 42, resignation is another blow to the prime minister hopes of getting the deal through the House of Commons.
May is facing a political firestorm for her plan to take the United Kingdom out of the bloc, including from within her own Tory Party.
If every opposition MP voted against, and they were joined by the 18 Tory MPs who have said they will also oppose it, this alone would be enough to defeat the deal.
The International Trade Secretary will visit the South West as he becomes the latest Cabinet minister to come out to bat for the Prime Minister ahead of the meaningful vote on December 11. "We will be explaining to Members of Parliament why we believe that this is a good deal for the UK", Mrs May told reporters on Thursday on the plane to the Group of 20 summit in Argentina when asked if she had a back-up plan.
Conservative MP Jacob Rees-Mogg, a staunch Brexiter who is one of the most vocal critics of the deal, said Friday that the government was trying to frighten people into accepting it with dire forecasts about the impact of leaving the European Union without an agreement.
"We would simply find ourselves in a period of more uncertainty and division", she said, adding: "It is absolutely important that we deliver on the vote that people gave".
"I think it is important members of parliament focus on the nature of this vote".
The Government on Wednesday published an assessment of the economic impact of Brexit, showing Britain would be worse off in any scenario outside the EU.
May said that if her plan was voted down by parliament the government and businesses would have to make decisions about implementing preparations for no deal.
Parliament will begin five days of debate on Dec 4, with the final vote due a week later. Instead, what I see from Labor is an attempt to frustrate what the government is doing to deliver Brexit for the British people.
And Straw highlighted a proposed amendment to the draft Withdrawal Agreement, submitted by a cross-party group of Remain-backing MPs, but with Labour support, as a step towards that. The EU has made clear there is little appetite to reopen negotiations.
Pro-EU politicians hope rejection of the deal could pave the way for a second referendum.
Mr Brown said: "Such actions on the part of the BBC are clearly not in line with the obligation to act fairly and leave the corporation open to the clear impression that you are acting on behalf of the Prime Minister's office, a position I am sure the BBC would not wish to be in".
Mr Tusk said in Buenos Aires Friday morning: "Many leaders will certainly have questions about Brexit, therefore let me say this".
She added: "The view I have had from farmers, employers and sector organisations and others in my trips around the country is that they recognise the importance of this deal and they support this deal".