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 at the end of October.
The latest candidate to announce his bid, Kit Malthouse, said: “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.”
The deadline the EU has set for Brexit is 31 October.
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.
avid to run for Tory party leader
Speaking to BBC Radio 4’s Today programme, Mr 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 Conservative 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, he said he had always believed that no deal was better than no Brexit, but warned that a prime minister advocating no deal would risk losing a confidence vote in Parliament – thereby effectively committing to a general election.
In that event, the Tories would mostly likely be “annihilated” and “face extinction” at the ballot box, he added.
“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”.
Tory backbencher and chair of the ERG Jacob Rees-Mogg said Mr Hunt’s proposal for a new negotiation team was a “very well intentioned offer”, but “the history of ministries of all the talents has not been hugely successful” and there may not be time to put together such a group before 31 October.
Mr Rees-Mogg, who supports leaving without a deal, also said any Tories who went against the government for pursuing his preferred option would be “excommunicating themselves from the Conservative Party” and be “putting Jeremy Corbyn into office”.
Housing Minister Mr Malthouse – who voted Leave -announced his candidacy in the Sun newspaper.
Speaking to the BBC’s Victoria Derbyshire programme, he said he would like to open up conversations with the EU to “get some movement on the withdrawal agreement or a new withdrawal agreement”, so the UK can leave with a deal.
But he said it would be the bloc’s choice whether they refuse to do that, thereby “effectively choosing no deal on our behalf”.
Mr Malthouse added: “My view is we need to get ourselves in as good a shape as we possibly can by 31 October – either by landing a deal, which is what the vast majority of MPs want in the House, or that we are in a coherent shape for no deal, so if the EU elect to go for that, we are able to go over the line as smoothly as we possibly can, not withstanding the fact it is is going to be bumpy.”