Justice Secretary David Gauke says he will resign if the next prime minister chooses to pursue a no-deal Brexit.
Tory leadership favourite Boris Johnson has pledged the UK will leave the EU on 31 October - with or without a deal.
However, Mr Gauke told the BBC's Andrew Marr Show that a "sizeable" number of Conservative MPs believed the UK should leave with a deal.
His comments come as Tory MP Sam Gyimah said more than 30 Tory MPs could vote against a no-deal Brexit.
The EU has set the UK a deadline of 31 October to leave the bloc.
Mr Gauke said he believed Parliament "will find a mechanism" between now and 31 October to prevent the UK leaving without a deal.
When asked whether he thought he would be sacked from the cabinet if Mr Johnson became prime minister, he said: "I suspect that I will possibly have gone before then."
He added: "Assuming that he wins, if Boris's position is that he is going to require every member of the cabinet to sign up to being prepared to leave without a deal on 31 October, to be fair to him I can't support that policy - so I would resign in advance."
Former Tory leadership hopeful Mr Gyimah - who resigned as a minister over Theresa May's Brexit plan - said there were more than 30 Tory MPs looking at legislative options to block a no-deal Brexit.
He told Sky News: "I wouldn't want to announce them before they have been tested as being viable."
"But there is a real concern. The real concern here is not about Leavers or Remainers. The real concern here, is that this is not in the interest of our country."
He added: "What all this is about is staving off economic mayhem."
Pro-Remain Tory MP Dominic Grieve has suggested MPs could use a Commons vote on Northern Ireland on Monday to launch a fresh bid to block a no-deal Brexit.
The government has tabled a Bill to delay any new election to the Northern Ireland Assembly while talks to restore power-sharing are ongoing.
Northern Ireland has been without a functioning government since 2017, when the power-sharing parties split in a bitter row.
Mr Grieve told Radio 5 Live's Pienaar's Politics: "The chances are, if Brexit goes through - a no-deal Brexit - it is going to be the end of Northern Ireland's union with the United Kingdom, with serious political consequences flowing from it.
"That's a Bill that is a perfectly legitimate place to start looking at how one might make sure no-deal Brexits are fully debated before they take place."
Asked about the possible number of MPs who might back such a bid, Mr Grieve said he did not know.
He added: "Like all these things, colleagues are pulled in different directions, perfectly understandably, by various considerations."
Leader of the pro-Brexit European Research Group, Jacob Rees-Mogg, told 5 Live he thought the only way to stop no-deal was to pass a new law.
He added that he would be "very surprised" if that happened.
Mr Johnson has insisted he is not bluffing over his promise to stick to the 31 October deadline for leaving the EU - even if that means walking away without a deal.
Asked in an interview with the Sunday Telegraph if his commitment to 31 October was a bluff, Mr Johnson said: "No ... honestly. Come on. We've got to show a bit more gumption about this."
He added: "It's vital that our partners see that. They have to look deep into our eyes and think 'my god, these Brits actually are going to leave. And they're going to leave on those terms'."
His leadership rival Jeremy Hunt has also said he was willing to leave without a deal, although he told the Sunday Telegraph it was "not the most secure way of guaranteeing Brexit" because MPs would try to block it.
Mr Johnson and Mr Hunt have been travelling around the country as they seek to win backing from Conservative party members, ahead of the vote closing on 22 July.
Home Secretary Sajid Javid has come out in support of Mr Johnson, saying the former foreign secretary was "better placed" than Mr Hunt to "deliver what we need to do at this critical time".
Tory MP Mr Rees-Mogg has suggested Mr Javid - along with Chief Secretary to the Treasury Liz Truss - are the main candidates to become the next chancellor.
Mr Rees-Mogg, who is supporting Mr Johnson in the leadership contest, said both had "very strong" credentials.