Jeremy Hunt has said the next prime minister should be someone who is trustworthy, otherwise the UK risks a general election and no Brexit.
In an interview with the BBC, the foreign secretary said the ability to negotiate a new Brexit deal "is about the personality of the prime minister".
"I do believe I can be trusted to deliver this deal," he said.
Mr Hunt said he would leave the EU without a deal, but not if there was a "prospect of a better deal".
He and Boris Johnson are competing to take over from Theresa May, with the winner of the contest due to enter Number 10 on 24 July.
'Who, not how'
In an interview with BBC political editor Laura Kuenssberg, Mr Hunt said he and Mr Johnson both wanted to change the Brexit deal negotiated by Mrs May - and their aspirations for the substance of a new deal were similar.
But, he said: "The judgement is - who is the person we trust as prime minister to go to Brussels and bring back that deal?
"If you choose someone where there is no trust, there's going to be no negotiation, no deal and quite possibly a general election - which can mean we have no Brexit either."
Mrs May's deal has been rejected three times by MPs, but Mr Hunt said he believed that there was a better "deal to be done" with Brussels and he was the man to get it.
"We've had a lot of discussion about how, but we need to have more of a discussion about who."
When asked whether he was implying that rival Mr Johnson was untrustworthy, he said he would "never" make those comments about a colleague.
"No, I'm saying I am trustworthy and I believe that I can be trusted to deliver that deal," he added.
He said if Mr Johnson became prime minister he would serve him to the "best of my ability" and he hoped Mr Johnson "would do the same as me".
Mr Johnson told the BBC on Monday that anyone questioning his character was "talking absolute nonsense".
At the moment the UK is leaving the EU on 31 October, after the date was twice extended from 29 March.
Mr Johnson has insisted he will leave by that date "deal or no deal" and challenged his opponent to do the same.
But Mr Hunt called the date a "fake deadline" that could "trip us into a general election and hand the keys to Jeremy Corbyn and we will have no Brexit at all".
He said that while he was prepared to leave without a deal, he would not pursue a no-deal Brexit if there was "a prospect of a better deal" within reach.
"What Theresa May tried to do was a deal involving the backstop," he said, referring to the controversial policy that aims to avoid a hard border on the island of Ireland.
"I was in cabinet at the time and I supported her loyally, but I never thought that was the right approach.
"What I'm talking about is a deal that doesn't involve the backstop as it's constituted, so it would be different."
When asked how he would solve the issue of the Irish border without the backstop, he said he would pursue a "technology-led solution".
He said he believed the government would "know very soon, well before 31 October" whether a new deal could be negotiated, and based on that, he would make a decision about whether or not to pursue no deal instead.
The EU has repeatedly said it will not renegotiate the withdrawal agreement, and has rejected the idea that technology could overcome the potential Irish border issues.
Jeremy Hunt is the underdog in this race and where does he see an opportunity to change that? Well, by talking about character.
He knows very well that many Tory members, many MPs, many members of the public, have real doubts about whether or not Boris Johnson is suitable to take the reins at Number 10.
And although Mr Hunt tried very hard to deny it - he said he was not suggesting Mr Johnson was actually untrustworthy - it was very telling that he chose again and again to emphasise he is the one to trust - and only if someone you can trust is in charge is there a possibility of getting a revised Brexit deal.
He clearly is determined to push and push again at the doubts that are in some people's minds about the suitability, the temperament and the character of the frontrunner in this race.
Clearly, they are very different political characters - one the showman, one wanting to pitch himself as the steady hand.
But it's also important to understand we should not let that mask the fact that there isn't a vast amount of difference between what they are both promising on the EU.
And both men's proposals for how they get us out of the Brexit mess are based on suggestions the European Union has turned down on many occasions.
Mr Hunt also set out his policy on social care in the interview, suggesting a system of automatic contributions towards costs where people are rewarded for "doing the right thing".
He said councils did need more money for social care and people wanted to know that as they get older "they're going to be properly looked after".
"So I think there is a bit of public money, but it's also about personal responsibility," he said.
"I think we should be a country where people save for their social care costs, particularly those last few months, possibly years, of their life when things can be very uncomfortable, very painful, just in the same way they save for their pension.
"And I think it can be something that people can opt out of, but it should be an automatic thing."
Views on abortion
The Tory leadership contender was also quizzed on his view that the legal time limit for an abortion should be reduced from 24 weeks to 12.
He said there were "very strong views" on "all sides" of the debate, but as health secretary he "did not seek to change the law" and it "will be the same as prime minister".
He said if there was a vote on legal time limits brought to Parliament, it would be "a matter of conscience".
On his ambitions to be prime minister, he said he had been "waiting for this moment for 30 years of my life".
"I have been sitting around that cabinet table thinking how I want to transform our country," he said.
"I think this is a moment, when I look at Brexit and this incredible moment in our history, when we could really unleash our potential and that's what gets me up in the morning."
Asked what it was, 30 years ago, that inspired him, he joked: "I won't say that. I think if I say that it's really going to put people off.
"But look, I would love to do this job, I think I can make a difference.
"I think this is a moment where if we send the right person, we can't just solve Brexit, we can open a whole new chapter in our history, an exciting chapter. That's what I want to do."
A debate on Sky News between the two leadership hopefuls was cancelled, after Mr Johnson declined to take part. Mr Hunt instead used the time on Monday evening for a Q&A on Twitter.
Replying to a question on student loans he said he would look again at the repayments process adding he could not "justify 6% interest rates". He also said that in housing policy he would introduce a "Right to Own" scheme aimed at helping young people to buy a house.
The cabinet minister used the question session to display a lighter side, comparing himself to 1970s sitcom character Basil Fawlty and joking that he was getting "useful ideas" from the notoriously brutal TV show Game of Thrones.