Tory leadership candidate Jeremy Hunt warns 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.
One leadership contender, Esther McVey, said "political suicide" would be not leaving the EU at the end of October.
The deadline the EU has set for Brexit is October 31.
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, Hunt said the "only solution" to deliver Brexit was to change the withdrawal agreement that Prime Minister Theresa May has negotiated with the EU, which has been voted down by MPs three times.
The EU has consistently refused to reopen the negotiations, but the foreign secretary said creating a new UK negotiating team - including members of Northern Ireland's DUP and members of the Tory pro-Brexit European Research Group, as well as Tory representatives from Scotland and Wales - would "give the EU the confidence that any offer can be delivered through Parliament".
He claimed the main issue that needed addressing in the withdrawal agreement was the Irish backstop - the policy to prevent a hard border returning between Ireland and Northern Ireland that has proved controversial with a number of Brexiteers.
"If you go to [the EU] with the right tone, with a deal that recognises their legitimate concerns over the border of Ireland, if you go with a negotiating team that gives them the confidence you can deliver with a majority in the House of Commons, you have the prospect - I don't say the guarantee - but you have the prospect of getting a deal," he said.
Writing in The Daily Telegraph, Hunt also warned the Conservatives would be "annihilated" and "face extinction" if there was a general election before Brexit happened.
He says he has always believed that no-deal is better than no Brexit but warns that a prime minister advocating no-deal would risk losing a confidence vote in Parliament - so committing to a general election.
"Trying to deliver no deal through a general election is not a solution; it is political suicide," he writes, adding it would "probably put Jeremy Corbyn in No 10 by Christmas".
New contender joins race
Housing Minister Kit Malthouse has become the latest Tory MP to join the race to become party leader.
Writing in the Sun newspaper, Malthouse said the campaign "cannot be about the same old faces" and described himself as "the new face, with fresh new ideas".
He told BBC News he was "the only candidate that has proven the ability to unify MPs around a Brexit plan that could deliver us out of this jam", after he brought together Leave and Remain-supporting Conservative MPs in January to devise an alternative Brexit proposal known as the "Malthouse Compromise".
Malthouse - who voted to leave the EU - said his "primary objective" was to secure a deal before the 31 October deadline, but said it was still necessary to prepare for the possibility of no deal.
Meanwhile, fellow leadership-hopeful Mr Gove has pledged to allow EU nationals in the UK at the time of the referendum to apply for citizenship free of charge.
He has said he wants to give as many as three million people from elsewhere in the EU who live in the UK an easier path to citizenship after Brexit.
As part of the plan, he would remove the requirement of EU citizens 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, Tory rival, International Development Secretary Rory Stewart, is promising a "listening exercise" on Brexit.
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 Tory leadership.
The declared candidates to replace 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