Leaving the EU by the end of October is a "hard red line" and will happen in "all circumstances", Andrea Leadsom has said in her pitch for leadership.
The ex-Commons leader said she had a plan for a "managed exit", adding that Parliament could "not stop us leaving".
But her rival Mark Harper said it was "not possible" to leave by 31 October, and Rory Stewart said talk of a better deal on the table was a "fairy story".
Ten Conservative candidates are in the race to be leader - and the next PM.
The deadline for Brexit was pushed back to October after MPs rejected Theresa May's withdrawal agreement with Brussels three times.
The European Union has repeatedly said the agreement will not be re-opened, and on Tuesday, president of the European Commission Jean-Claude Juncker stressed that once again.
"This is not a treaty between Theresa May and Juncker, this is a treaty between the United Kingdom and the European Union," he told a Politico event in Brussels.
"It has to be respected by whomsoever will be the next British prime minister."
Irish Prime Minister Leo Varadkar said it was a "terrible political miscalculation" for UK politicians to believe they can get a better deal.
What have the leadership contenders said about Brexit?
"In all circumstances we are leaving the European Union on 31 October," Mrs Leadsom told her official campaign launch. "Our country and our party cannot afford any more indecisiveness."
The Brexiteer MP set out her plan for what she calls a "managed exit" from the EU, which includes striking a "temporary trade agreement" and a plan to negotiate contingency arrangements with Brussels over the summer recess.
She said these could be discussed at a summit with the new incoming EU commissioners and heads of government in September.
But at his official campaign launch, Mr Harper - an outsider in the race - said it was "not possible or credible" to leave on the terms of a new deal by the existing deadline of 31 October. Renegotiating and getting a deal past MPs would take longer, he said.
He said there could be a majority in the Commons to leave without a deal, but only if ministers demonstrated they had "strained every sinew" to get a new one.
Meanwhile, Health Secretary Mr Hancock - who is also competing for the top job - told BBC Radio 4's Today programme his plan was "eminently deliverable" by 31 October, as the EU was open to changing the political declaration part of the agreement.
"We need to solve Brexit and we cannot do it by threatening no deal," he said, adding: "Parliament will not allow a no-deal Brexit to happen."
Home Secretary Sajid Javid reiterated that although he wanted a revised deal, "if we got to end of October and the choice was between no deal or no Brexit, I'd pick no deal."
Later in the day, launching his campaign in a circus tent in London, Rory Stewart - another outsider - said he wanted to "take the politics out" of the situation and find a way to get Mrs May's deal through Parliament.
He said he would ramp up pressure on MPs to back it by promising that otherwise, he would convene a "grand jury" of citizens to make recommendations on how to proceed which politicians would have to stick to.
Vowing to hold the UK together and reconcile "extreme Remain and extreme Brexit" arguments, he added: "My project is about one thing - it is about moderation and compromise."
What else have the leadership candidates said?
At her official launch, Mrs Leadsom introduced several policies away from Brexit, including using overseas development aid to help poorer countries to decarbonise and helping young people to save for a house deposit with a new scheme.
When questioned on taxes - prompted by Boris Johnson's pledge to cut income tax for those who earn more than £50,000 a year - she said she believed in low taxes.
But tax reform could not get through a hung Parliament, so that "needs to wait".
Taking a swipe at Mr Johnson's idea, Mr Harper said he would focus his tax cuts "at the lower end of the spectrum", adding: "I don't think we should be promising more money to higher rate taxpayers."
He also said the lack of a majority meant certain things would not be deliverable and as Brexit showed, it was "toxic" to make promises and not fulfil them.
Who will replace Theresa May?
The winner of the leadership contest will become next Conservative leader and prime minister. They're due to be in place by the week beginning 22 July.
Mr Javid, who launches his campaign on Wednesday, released a campaign video which BBC political editor Laura Kuenssberg described as the first big attempt by a candidate to communicate a personal story, introducing viewers to his family and background.
He also told the Evening Standard he was "very open minded" about having different immigration rules for regions such as London after Brexit, and could scrap Mrs May's policy that EU migrants should earn at least £30,000 to be considered for admission.
Mr Johnson - accused by Michael Gove and other candidates of "hiding in his bunker" because he is yet to a do a major event or TV interview - is also launching his campaign on Wednesday.
It's day two of the official campaign to be the next prime minister.
Andrea Leadsom cheerily launched her campaign, promising she would never utter the phrase "as a mother" that did for her chances last time.
As promised, the former chief whip Mark Harper was jacket off, sleeves rolled up, answering any question that journalists were willing to put.
That included - because the early stages of this campaign are this surreal - predicting in a fight between a lion and a bear that the lion, patriotically, would win. (yes, you read that right).
And TV presenter Lorraine Kelly was back - this time with a slapdown of the whole lot of the political class.
But the hard reality bites today too. Labour has just announced that they are leading another cross-party attempt to grab control of the Commons.
Just in case the candidates needed a reminder of what they'll inherit, the politician who wins this race might find that MPs have changed the law to kill off their solution to Brexit before they even call the removal vans to move their stuff to Number 10.
How does the contest work?
Ten Conservative candidates will contest Thursday's first round of voting after nominations closed in the contest to succeed Theresa May as Tory leader and prime minister.
Over the next two weeks, Tory MPs will take part in a series of secret ballots to whittle the candidates down to the final two.
The party's 160,000 or so members will then pick a winner in a postal ballot, with the result announced in the penultimate week of July.
On Tuesday 18 June BBC One will host a live election debate between the Conservative MPs still in the race.
If you would like to ask the candidates a question live on air, use the form below. It should be open to all of them, not a specific politician.
If you are reading this page on the BBC News app, you will need to visit the mobile version of the BBC website to submit your question on this topic.