Tory leadership contender Jeremy Hunt has refused to guarantee that the UK will leave the EU before Christmas, but said he "expects" it to happen by then.
He would not say when Brexit would take place if he became PM, telling the BBC: "I'm being honest with people".
Rival Boris Johnson said the UK would leave by 31 October "come what may".
He also defended his remarks on the UK ambassador in Washington, who quit this week over leaked criticisms of Donald Trump.
Mr Johnson added he did not accept that his failure to support Sir Kim Darroch during a debate on ITV earlier this week had prompted him to resign.
However, he said a "misrepresented" account of his remarks later relayed to Sir Kim had been "a factor" in his decision to step down.
He added: "I stood up completely for the principle that civil servants should be allowed to say what they want to their political masters."
Up to 160,000 Conservative Party members are voting for their next party leader - and UK prime minister - to replace Theresa May.
The BBC's Andrew Neil has interviewed both contenders for a programme broadcast on BBC One.
Mr Johnson, a former foreign secretary and mayor of London, is seen as the frontrunner in the contest.
Mr Hunt warned party members not to "vote with their hearts instead of their heads".
He added that the "quickest way" to leave the EU was "to send to Brussels a prime minister who can negotiate a deal that will get through Parliament - and I'm that person".
Foreign Secretary Mr Hunt, who set up his own business before entering politics, was challenged on whether he had the skills to negotiate effectively with the EU.
He replied that being an entrepreneur had given him the "basics", adding: "In government those same skills I used to negotiate very complex things - like the licence fee deal with the BBC, the NHS pay awards, the protracted dispute to try and get a peace process going in Yemen - that business of negotiation is something I have been doing all my life."
Mr Hunt said the main change he wanted to see to the UK's current withdrawal deal was to the Irish border backstop plan - an insurance policy which aims to guarantee there will not be a hard border on the island of Ireland after Brexit.
He added that changes to this part of the Brexit withdrawal deal - which has been rejected three times by MPs - would "broadly" make it acceptable to the Commons.
When pushed on what else he would alter, Mr Hunt said that "there may be other elements", but did not provide further details.
On Parliament's attempts to block a no-deal Brexit, he warned that the UK needed to be "careful" about the 31 October deadline, and said: "I think I'm the best person to get a deal… but I can't control what Parliament does."
Asked whether Brexit would have happened by Christmas, Mr Hunt said: "I expect so."
He was then challenged on whether the UK would still be a member of the EU going into 2020, replying: "I don't believe so."
In his interview with Andrew Neil, Mr Johnson said he believed the UK would leave the EU on 31 October, and that if this did not happen it would lead to "a huge erosion of trust in politics".
"I think it is very odd that those who say they would delay even further can't set another date - I mean, how much further are we going to wait?" he said.
"I think it's very, very important that we get ready to leave on 31 October, come what may, and we will."
Mr Johnson said he did not want to prorogue - suspend - Parliament to push a no-deal Brexit through, but he would not rule it out.
The UK's ambassador in Washington, Sir Kim Darroch, resigned on Wednesday after a row over leaked emails critical of President Donald Trump's administration.
Mr Johnson was criticised in the aftermath for failing to fully support Sir Kim in the ITV leadership debate the evening before. This followed angry criticism of Sir Kim by Mr Trump.
Mr Johnson said he had spoken to Sir Kim on Thursday to express his sadness over his resignation and the ambassador told him he had not watched the TV debate.
But Mr Hunt said: "I think we have to back our diplomats all over the world.
"Sir Kim was doing his job. He was giving his own personal but totally honest view about the country he was serving in."
Tax and spending pledges
On economic policy, Mr Hunt admitted that some of his spending pledges would take longer to deliver if the UK left the EU without a deal.
But he insisted that even in a no-deal scenario, he would push ahead with his plan to cut corporation tax - adding it would help firms cope with the resulting "shock" to the economy.
When asked whether he would continue with the current government's self-imposed limits on borrowing, Mr Johnson pledged to "continue to bear down on our national debt".
"We will be setting out in a Budget and a spending review exactly what we will be doing on the fiscal rules and everything else," he added.
The result of the Conservative leadership contest will be announced on 23 July, with the winning candidate taking over from Mrs May on 24 July.