Hunt told BBC Radio 4's Today programme on Tuesday that the move to no-deal would result in a general election, which could see opposition Labour Party take power.
He said the "only solution" to deliver Brexit was to change the withdrawal agreement that outgoing Prime Minister Theresa May negotiated with the EU, which was voted down by MPs three times.
The official leadership race for Conservative party leader gets under way in early June, after May stands down - but jostling between candidates has already begun. The winner, expected to be named by late July, will also become Prime Minister.
The EU has consistently refused to reopen the negotiations, but Hunt said creating a new UK negotiating team including members of Northern Ireland's DUP, Tory pro-Brexit European Research Group as well as 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 recognizes 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... of getting a deal."
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.
Housing Minister Kit Malthouse has become the latest Tory MP to join the race to become party leader.
So far, the candidates vying to replace May are: Hunt, former Foreign Secretary Boris Johnson, Environment Secretary Michael Gove, Health Secretary Matt Hancock, Home Secretary Sajid Javid, former Leader of the House Andrea Leadsom, Housing Minister Kit Malthouse, former Work and Pensions Secretary Esther McVey, former Brexit Secretary Dominic Raab and International Development Secretary Rory Stewart.
( With inputs from IANS )