Boris Johnson has said the British people are “fed up” with the Brexit deadlock and the Tories will pay a “serious price” if the UK fails to leave the EU on 31 October.
In a live BBC TV debate, the former foreign secretary said there would be a “catastrophic loss of confidence in politics” if the deadline was not met.
Sajid Javid said it was “fundamental” that the UK left the EU by this point.
But Michael Gove and Jeremy Hunt said they could seek extra time if needed.
And Rory Stewart said it was not possible to negotiate a new deal.
He suggested he was seeking the most realistic “door” out of the EU while “everyone else was staring at the wall and saying believe in Britain”.
Asked whether he could guarantee that the UK would leave the EU by 31 October, Mr Johnson said it was “eminently feasible”.
The five men vying to be Conservative Party leader – and the UK’s next prime minister – are taking part in a live televised debate on BBC One.
Mr Johnson, Mr Hunt, Mr Gove, Mr Stewart and Mr Javid are answering questions from viewers via video links from BBC studios.
There is no studio audience and candidates are not making opening or closing statements.
Our Next Prime Minister is being hosted by Newsnight presenter Emily Maitlis.
Former Brexit Secretary Dominic Raab was earlier eliminated in the second round of voting, when Conservative MPs voted by secret ballot in the latest stage of the contest.
The BBC received thousands of questions ahead of the debate, with the largest number being about Brexit. Other topics submitted included HS2, public spending and climate.
Each candidate will be given an opportunity to answer every question posed.
The programme team selected the questions to be asked after making an editorial decision on the ones that best represent the issues likely to face the next prime minister.
The BBC said the questions will come from a cross-section of society, not necessarily all Conservative voters.
How can you watch the debate tonight?
Our Next Prime Minister is being hosted by BBC Newsnight presenter Emily Maitlis and broadcast on BBC One at 20:00 BST.
You can also follow the debate on our live page on the BBC News website.
Alternatively, you can stream the debate online through BBC Sounds or by watching BBC One and the News Channel through iPlayer.
From 21:00 BST, the BBC News Channel will be broadcasting reaction to the debate from politicians and pundits.
The debate is also being broadcast live on Radio 4.