Justice Secretary David Gauke has said he would be "very surprised" if the prime minister was prepared to back a no-deal Brexit, if her deal fails.
Mr Gauke also said he would find it "very difficult" to stay in cabinet, if that became government policy.
Cabinet splits have emerged over what should happen if the PM's withdrawal deal is rejected by MPs next month.
Andrea Leadsom has suggested a "managed no deal" Brexit while Amber Rudd said a new referendum was "plausible".
Mr Gauke supported Remain in the 2016 referendum.
Labour MP Chuka Umunna, who campaigns for another EU referendum, said the justice secretary had "effectively admitted that all the talk of leaving the EU without a deal is nonsense and a false threat designed to scare MPs into voting for the government's Brexit plan".
He added: "At a time when our schools, hospitals and police are desperately underfunded, the £4.2bn being spent preparing for Brexit would be far better spent on our public services."
The UK is due to leave the EU on 29 March 2019 but an agreement on the terms of its withdrawal and a declaration on future relations will only come into force if the UK and EU Parliaments approve it.
The Commons vote was due to be held earlier this month but Theresa May postponed it, once it became clear it would be defeated by a large margin.
MPs are due to start debating the deal again on Wednesday 9 January.
If Mrs May's deal is rejected, the default position is for the UK to leave in March unless the government seeks to extend the Article 50 negotiating process or Parliament intervenes to stop it happening.
Mr Gauke, who was interviewed on the BBC's Political Thinking with Nick Robinson podcast, was asked about comments he reportedly made at a cabinet meeting on Tuesday, in which he dismissed the idea of a "managed no deal" Brexit as a "unicorn".
He told the BBC: "I think making a conscious decision to proceed with no deal would not be the responsible course of action and I think we would have to look at what other choices were available to us."
Asked if he would remain in the cabinet, he said: "I think it would be very difficult for me in those circumstances."
He said there was a risk of an "accidental no deal" and that "the best way of stopping no deal is to back the prime minister's deal, in my view".
But Mr Gauke said: "I think if it came down to the government saying, consciously, we will just have to do that, I don't think there would be a lot of support for it. I would be very surprised if the prime minister would be prepared to go down that route."
Mrs May has suggested the choice facing MPs is one between her deal, no deal or potentially, no Brexit at all. She has refused to rule out a "no deal" Brexit - under pressure from many MPs to do so.
On Thursday, Commons Leader Andrea Leadsom told the BBC a "managed" no deal did not have to mean no withdrawal agreement at all.
The Leave campaigner said it could be a stripped-down agreement incorporating some of the EU's no-deal preparations.
"What I am looking at is trying to find an alternative so that in the event that we cannot agree to this deal there could be a further deal that looks at a more minimalist approach but enables us to leave with some kind of implementation period.
"That avoids a cliff edge, that avoids uncertainty for businesses and travellers and so on."