Foreign Secretary Jeremy Hunt has warned that if the Conservative Party was to face a general election in the next six months then "we would be wiped out as a party".
Speaking on ITV's Peston, Mr Hunt also refused to rule out a further delay to Brexit beyond October 31, and insisted that a new deal that would be passed by Parliament could be negotiated with the EU before Halloween.
The former health secretary said the major sticking point over the current deal was the issue to the Northern Ireland backstop, and a deal to change or remove this could be negotiated with Brussels, even though the terms of the October 31 Brexit extension rule out changing the Withdrawal Agreement.
"A deal is a deal unless both sides decide to do it differently," Mr Hunt told ITV News' Political Editor Robert Peston.
"In this situation, you have to ask yourself - with a new prime minister, with a fresh mandate - whether, ultimately there is a deal there."
Following Prime Minister Theresa May's announcement that she will stand down on June 7 and a Tory leadership race will ensue, Mr Hunt is one of 11 candidates who have said they would like to be the Party's next leader.
While Mr Hunt praised Mrs May for "working incredibly hard" to get her Brexit deal, he said a new prime minister might be able to negotiate a new one.
Mr Hunt said that as prime minister he would seek a new deal with the European Union, and "the reason I think there is a deal there is because it's in both sides' interests to solve this: I think the EU don't want the shadow of Brexit hanging over them, they want this issue solved."
He said that "one of the reasons that we didn't get the flexibility we wanted from the EU was because they lost faith that the British government would be able to deliver the British Parliament" so the negotiating team should have representatives from the DUP and Tories from the European Research Group.
"I don't pretend it's going to be easy," he acknowledged.
While Mr Hunt said no-deal was preferable to no Brexit, he refused to rule out a further delay to Brexit beyond October 31.
"I believe it can be done (by October 31)," he told ITV's Peston, but warned that no-deal could result in a general election, something he did not believe the Tories would be able to weather.
"What I would say to the other candidates is 'if you are saying you will leave on October 31, deal or no deal, then are you - if Parliament stops you - prepared to have a general election to change Parliament?
"My commitment is that I would not, and I would challenge all of the candidates to say exactly where they stand on that because I think people need to know whether there is a risk of a general election in the next six months and ... I think that would be very, very devastating."
He added: "I think what most people will be thinking is 'who is most likely to deliver Brexit and deliver it quickly'.
"If we get the approach wrong, if you send an ultra hardliner to Brussels you will get an ultra hardline response. And then we will be faced with those very difficult choices."
Also on Peston was fellow leadership hopeful James Cleverly, who unlike Mr Hunt advocated more freely for no-deal, saying it was better than the "indecision" the country currently faces.
"We have got to deliver Brexit," Mr Cleverly told Peston.
"Delay is the worst thing... delay brings indecision... we've got to get real and make the tough decisions," he said.
The Braintree MP continued that delay over Brexit and indecision over a deal is "killing" businesses, and said that even business owners who voted remain would rather leave the EU with no-deal as it is preferable to delay.
While Mr Hunt portrayed himself as a prime minister who would "negotiate" with the EU, pointing to his past in business and many years spent in the Cabinet, Mr Cleverly painted himself as a political relative outsider, having only become an MP in 2015.
"If you look at global politics, there is a pattern emerging," the 49-year-old said.
He continued that voters are rejecting politicians who have been around for "years and years and years" and who are "stale" and are instead voting for those who "look and sound different" and offer a new approach.
Should he become prime minister Mr Cleverly said he would "get Brexit off the table" before focusing on anything else.
While both Mr Cleverly and Mr Hunt said they would be prepared (to varying degrees) to back no-deal, Labour MP Ben Bradshaw said the idea that Parliament would allow this to happen was "for the birds", and it would find a way to "frustrate" this, since the current majority backs leaving with a deal.
Meanwhile on the BBC, leadership hopeful Matt Hancock distanced himself from Mrs May's "no-deal is better than a bad deal" mantra and insisted he would be "brutally honest" about the difficulties of the process.
Mr Hancock said he is "not against" a no-deal Brexit but added: "It won't happen because Parliament won't allow it", echoing Mr Bradshaw's comments to Robert Peston.
The "best option" was to secure a deal and he told Newsnight that would require the backstop - measures which keep the UK closely bound to EU trade rules to avoid a hard border with Ireland - to be replaced, comments similar to those made by Mr Hunt.
Asked what he could do differently to Mrs May, he said: "I'm being completely straightforward about the real trade-offs that exist.
"I haven't started in this saying that 'no deal is better than a bad deal'. I haven't started from the point of view of saying that.
"I've started from the point of view of saying we've got to be brutally honest about the real constraints that will face the next prime minister, whoever they are."
He said he would put "alternative arrangements" at the centre of his plan.
"The difference is that the alternative arrangements have not been at the forefront of the policy that was part of the deal put forward before.
"Instead the proposed future arrangements were to build on the backstop and that is materially different."