Richard Harrington, who serves in the Department for Business, Energy and Industrial Strategy, said he would "definitely quit" in such a scenario, but supported the prime minister's deal which MPs will vote on next week.
His remarks - immediately after a Downing Street drinks reception for Conservative MPs - followed a government-funded exercise aimed at testing the resilience of the transport network under a no-deal.
It also comes as MPs attempt to limit the government's financial powers and its ability to pursue further no-deal preparations, in a vote that is expected to demonstrate the strength of feeling against a disorderly exit from the EU in the Commons.
Speaking outside Downing Street, Mr Harrington told BBC's Newsnight that the UK was going to leave the bloc "with the prime minister's deal", claiming: "I think people are beginning to realise that it's the prime minister's deal or there may not be a Brexit.
Pressed on whether he would resign as a minister to prevent a no-deal scenario, he replied: "Definitely, I would."
"The prime minister knows everybody's views and I think my view is not an uncommon one. But we fully support the prime minister's deal and I think it will get through."
Theresa Villers, a former Tory cabinet minister, who also attended the drinks reception on Monday evening, said she had not changed her view on the prime minister's deal and will still vote down the UK-EU agreement.
And Nigel Dodds, the Westminster leader of the Democratic Unionist Party (DUP), which is responsible for propping up Ms May's fragile government, described the meeting as a "very pleasant social occasion".
"Let us wait and see what comes forth in the course of the next few days but there hasn't been much so far, so we'll see what happens in the next little while."
Ms May is expected to hold a second reception for MPs on Wednesday in a further attempt to persuade them to vote for her deal on the 17 January, following five days of debates in the Commons.
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