Tory leadership candidate Jeremy Hunt has warned that his party will be committing "political suicide" if it tries to push through a no-deal Brexit.
He told BBC Radio 4's Today programme the move to no deal would result in a general election, which could see Labour take power.
The foreign secretary is one of 10 people seeking to replace Theresa May.
Another contender, Esther McVey, said "political suicide" would be not leaving the EU on 31 October.
The UK's departure was pushed back to that date after the country missed the previous deadline of 29 March.
The official race to be Conservative Party leader gets under way in early June, after Theresa May stands down - but jostling between candidates has already begun.
The winner, expected to be named by late July, will also become prime minister.
Speaking to BBC Radio 4's Today programme, Mr Hunt said he wanted to change the withdrawal agreement that Mrs May negotiated with the EU - despite the bloc repeatedly refusing to re-open talks on the document.
He also pledged to create a new UK negotiating team - drawn from all sides of the Tory Party, plus members of Northern Ireland's DUP - to "give the EU the confidence that any offer can be delivered through Parliament".
Several leadership contenders, including Boris Johnson, have said they would be prepared to leave on 31 October without a deal with Brussels.
But, writing in the Daily Telegraph, Mr Hunt warned that a prime minister advocating that option would risk losing a confidence vote in Parliament - thereby effectively committing to a general election in which the Tories would mostly likely be "annihilated".
It would "probably put Jeremy Corbyn in No 10 by Christmas", he added.
Tory backbencher and leading Brexiteer Jacob Rees-Mogg said Mr Hunt's proposal for a new negotiating team was a "very well intentioned offer", but there may not be time to put together such a group before 31 October.
Mr Rees-Mogg also said any Tories prepared to vote against their own government for pursuing a no-deal exit must understand they would be "putting Jeremy Corbyn into office".
The latest candidate to announce his leadership bid, Housing Minister Kit Malthouse, told the BBC's Victoria Derbyshire programme he would like to "get some movement on the withdrawal agreement or a new withdrawal agreement" - and if a new PM approached Brussels "with the right tone" and negotiating team, there was "the prospect of getting a deal".
But he said the EU could refuse to play ball, thereby "effectively choosing no deal on our behalf".
And Mr Malthouse added: "Those people who say no deal would be a catastrophe and those people who say it would be a walk in the park are both wrong - it is somewhere in the middle."
Former Brexit Secretary Dominic Raab has also released a video, saying as leader he would focus on "fairness" - from cutting taxes for the lowest paid and lowering the cost of living, to increasing opportunities for young people.
"We live in an age when the actions of competent leaders and good government can and should go a long way to making the world a fairer place," he said.
"And that driving conviction, that things can and will be fairer, should be at the heart of what the future of the Conservative Party is all about."
Fellow hopeful Michael Gove has pledged to allow up to three million EU nationals in the UK at the time of the referendum an easier path to citizenship after Brexit.
As part of the plan, he would remove the requirement for them to provide proof of their right to be here - getting rid of the so-called "settled status" scheme.
A source close to the environment secretary said: "This is simply the right thing to do - honouring the promise of Vote Leave that EU nationals studying, working and living in the UK were welcome to stay."
Meanwhile, International Development Secretary Rory Stewart, is promising a "listening exercise" on Brexit if he wins the leadership race.
And the Sun reports that rival contender Health Secretary Matt Hancock has written to ITV, BBC, Sky and Channel 4 to ask them to broadcast a live debate between those vying for the job.
Who is running for the Tory leadership?
The declared candidates to replace Mrs May are:
- Environment Secretary Michael Gove
- Health Secretary Matt Hancock
- Foreign Secretary Jeremy Hunt
- Home Secretary Sajid Javid
- Former Foreign Secretary Boris Johnson
- Former Leader of the House Andrea Leadsom
- Housing minister Kit Malthouse
- Former Work and Pensions Secretary Esther McVey
- Former Brexit Secretary Dominic Raab
- International Development Secretary Rory Stewart