Meanwhile, Amber Rudd, the work and pensions secretary, ruled herself out of the contest and hinted she could sit in a cabinet with Mr Johnson, the clear favourite.
In contrast, Mr Stewart tore into the former foreign secretary after he vowed to crash the UK out of the EU without an agreement in October, if necessary.
“I could not serve in a government whose policy was to push this country into a no-deal Brexit. I could not serve with Boris Johnson,” he told the BBC.
Mr Stewart – a candidate himself – added that he had spoken with Mr Johnson only two weeks ago and believed “he had assured me that he wouldn’t push for a no-deal Brexit”.
Yet, within hours of the prime minister’s tear-stained resignation, he had ruled out a further Article 50 extension, telling an audience: “We will leave the EU on October 31, deal or no deal.”
“I spoke to Boris, I suppose, about two weeks ago about this and I thought at the time he had assured me that he wouldn’t push for a no-deal Brexit,” Mr Stewart said.
“So, we had a conversation about 20, 25 minutes and I left the room reassured by him that he wouldn’t do this. But, it now seems that he is coming out for a no-deal Brexit.”
Mr Hancock, the youngest candidate at just 40, said he was running for leader because the party needed to look to the future and attract younger voters.
Arguing against a crash-out Brexit, he vowed to pursue a different approach to try to get Commons support for a deal than the one Ms May failed with.
“She didn’t start by levelling with people about the trade-offs,” Mr Hancock said.
“I think it is much, much easier to bring people together behind a proposal if you are straightforward in advance.”
The health secretary also sought to woo Tory MPs terrified about the prospect of an early general election, insisting it should not happen before Brexit is achieved.
“Some of my contenders may say that, if they don’t get their preferred option, whether it be no deal or something else, then they’ll have a general election,” he said.
“I put it to you that would be a disaster for the country and it would risk Corbyn by Christmas.”
Liam Fox, the international trade secretary, refused to rule himself out as he warned a no-deal Brexit was “increasingly likely”.
Asked if he would stand, Mr Fox told the BBC: “Well, I don’t think it’s likely to happen, but, as you say, it’s an unusual contest.”
Arguing it was better to leave with an agreement, he added: “I think that it is possible, and probably increasingly likely now, that we could leave without a deal because I think there’s limited patience from the European Union with Britain’s constant delay.”
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