Both the frontrunner, Boris Johnson and Jeremy Hunt say they would be willing to take the UK out of the EU without a deal on 31 October, but could risk collapsing their government by seeking to push a no-deal Brexit through parliament.
As many as 30 Tory MPs are ready to vote against a no-deal scenario in parliament, with the government’s working majority now just three.
It comes as Mr Johnson has insisted he is not bluffing over his commitment to leave the EU on 31 October - with or without a deal.
The Tory leadership frontrunner said the EU had to "look deep into our eyes" and realise that the UK was prepared to walk away.
Asked in an interview with the Sunday Telegraph if his commitment to the 31 October deadline was a bluff, Mr Johnson said: "No ... honestly. Come on. We've got to show a but more gumption about this."
His campaign received the support of Home Secretary Sajid Javid, who said Mr Johnson was "better placed" than Jeremy Hunt to "deliver what we need to do at this critical time".
Mr Javid, who is widely believed to have his sights set on becoming chancellor under the new prime minister, called for an emergency budget for a no-deal Brexit.
"Trust in our democracy will be at stake if we don't make October 31 a 'deal or no deal' deadline,” he said.
"To prepare that, we are agreed on the need for ramped-up no-deal preparations, including a budget."
Mr Hunt also insisted he was not bluffing about being willing to walk away without a deal, although he told the Sunday Telegraph it was "not the most secure way of guaranteeing Brexit" because MPs would try to block it.
Former Tory leadership hopeful Sam Gyimah said more than 30 of the party's own MPs could vote against a no-deal Brexit.
Speaking on Sky News, the former education minister said there are a "significant number" of Tory MPs who are looking at legislative options to block a no-deal Brexit.
He added: "I think it is about 30, 30-plus. But what they will be looking to do is stop a no prime minister from proroguing Parliament in order to deliver no deal.
"But also create options for the new prime minister so that no deal is not the only option we face on October 31."
Mr Gymiah warned no-deal was “the most successful way to have a second Scottish independence referendum [and] break up the UK”.
He said he was not prepared to vote for a motion of no-confidence in the government over Brexit, but Mr Grieve confirmed he would do so “with great reluctance”.
Speaking on the BBC’s Pienaar’s politics programme, the former Attorney General said: “We are going to have in the course of the next twenty four hours an important bill on Northern Ireland. Northern Ireland and Brexit go very closely together.
“The chances are that if Brexit goes through as No Deal Brexit it’s going to be the end of the Northern Ireland union with the United Kingdom with serious political consequences flowing from it.
“So that’s a bill which is perfectly legitimate place to start looking at how one might make sure how a Brexit deal is fully debated before it takes place.”
Mr Gauke suggested that the “innovative” Commons speaker John Bercow would ensure that “parliament will find a way through” to block no-deal.
And the Justice Secretary said it would be “preposterous” for parliament to be prorogued to force through a no-deal Brexit, something that Mr Johnson has ruled out.