Liam Fox, the global trade minister, said Thursday the Cabinet could block May from bringing her exit withdrawal deal before the House of Commons next month and that it might insist Britain's scheduled departure date of March 29 be moved back several months to provide time for a fresh approach.
David Lidington, David Gauke, Amber Rudd, Philip Hammond and Greg Clark are understood to have stepped up discussions over whether to send the issue back to the public.
Mr Hunt's upbeat remarks on a no deal scenario come amid rising tensions in Mrs May's Brexit war cabinet.
Pointing to EU Council president Donald Tusk, a former Polish PM, the long-serving ex-premier of Luxembourg said: "The two of us were prime minister and we sometimes had to face motions asking for our resignation".
It will include demands for a major step-up in planning for a no-deal Brexit.
But she added: 'The EU is clear, as I am, that if we are going to leave with a deal, this is it. Linked to this notion of control is the politically fraught issue of immigration, with the influx of low-skilled workers from other parts of Europe, and the resulting pressures on wages and social services, providing the pretext for advocating an extremely parochial and often racist discourse on immigration that sees leaving the European Union as the principle means through which greater control could be exerted over the UK's borders.
In a pointed swipe at the Labour heavyweight, the Prime Minister said a second referendum would amount to Parliament abdicating responsibility.
He is under pressure to launch a confidence vote in the government in the Commons, but has held off so far because he believes Mrs May would win.
However, the Labour leadership has made it clear it will only strike when it considers the Government to be at its most vulnerable.
"So, we know that we would have delays at customs. It means that food will not be on shelves".
After listening to a presentation from the prime minister about MPs" concerns which had prevented her getting her Brexit deal through Parliament this week, the EU27 leaders tore up a draft communique prepared by officials which would have offered Mrs May "further assurances'.
Many Tory Brexiteers, and the Democratic Unionist Party (DUP) that Mrs May depends upon for a Commons majority, have rejected the backstop proposals.
Many of Mrs May's MPs are concerned that the "backstop" - which is aimed at preventing a hard border in Northern Ireland - would keep the United Kingdom tied to European Union rules and limit its ability to strike trade deals.
She told the press conference: "I had a robust discussion with Jean-Claude Juncker - I think that's the sort of discussion you're able to have when you have developed a working relationship and you work well together".
As Juncker puts his hand on May's arm to placate her, Dutch Prime Minister Mark Rutte is seen striding across the room to restore the peace.