Jeremy Hunt has urged Tory leadership rival Boris Johnson to "be straight with people" about what a no-deal Brexit would mean.
The foreign secretary said Mr Johnson's "million to one" claim about the chance of a no-deal "flies in the face of reality".
Mr Johnson said any suggestion Brexit could be delayed again would "end up eroding trust in politics".
Both men have said they would try to renegotiate a deal with the EU.
But Mr Johnson says the UK must leave the EU on 31 October, "do or die", with or without a deal.
Mr Hunt says he would leave without a deal in October if there was no prospect of leaving with one - but has not ruled out a further delay and has called 31 October a "fake deadline".
They are competing for Conservative Party members' votes in the race to replace Theresa May as Conservative leader - and prime minister.
At a hustings in Bournemouth, Mr Johnson was asked if he would rule out suspending Parliament - a controversial move - in order to push through a no-deal Brexit.
Describing it as an "archaic device", he said: "I'm not attracted to the idea of a no-deal exit from the EU but, you know, I think it would be absolutely folly to rule it out. I think it's an essential tool of our negotiation.
"I don't envisage the circumstances in which it will be necessary to prorogue Parliament, nor am I attracted to that expedient."
A no-deal exit would see the UK leave the customs union and single market overnight and start trading with the EU on World Trade Organization rules.
Mr Johnson has said a mechanism known as GATT 24 could be used to prevent tariffs, if there was a no-deal Brexit.
But in a letter to his rival, Mr Hunt quoted Leave-backing cabinet ministers Liam Fox and Geoffrey Cox, who have argued that this would require a deal with the EU.
In his letter, he asked: "Who is correct: You, or the Attorney General and the International Trade Secretary?"
He also questioned Mr Johnson's suggestion that a free trade agreement could be negotiated during an "implementation period", if no deal was reached - saying it was a "fact" that without a deal, there would be no implementation period and Brussels negotiators put the political cohesion of the EU before economics.
"We must be careful to face the facts as we find them. Will you be straight with people that no deal means no implementation period?"
At the hustings, Mr Johnson criticised Mr Hunt's suggestion that the current Brexit deadline of 31 October could be delayed again - having been pushed back from 29 March after MPs repeatedly rejected the deal Mrs May had agreed with the EU.
"Anybody who proposes any further delay is simply going to end up eroding trust in politics, eroding people's confidence in our democratic institutions further," he said.
"And further weakening out great Conservative Party and our mission to lead this country.
"And it simply won't work. Kick the can again and we kick the bucket, my friends, that's the sad reality."
He also stood by his suggestion that the chances of leaving the EU without a deal were a "million to one", arguing there had been a "change in mood in Westminster" and that there was now a "growing opportunity to get this thing done with style".
In other answers from the hustings, Mr Hunt suggested he would quit as PM if he failed to deliver Brexit. Asked if he would "fall on his sword", he said: "Of course, no PM is going to last any time at all if they don't deliver Brexit and deliver it very quickly."
He also ruled out involving Brexit Party leader Nigel Farage in negotiations with the EU: "Nigel Farage doesn't want a deal, he wants a WTO Brexit straight away.
"And while I would be prepared to do that if there was no other alternative and I'm absolutely clear about that, I think it would be much better for our businesses and much better for our Union if we could get a deal and I haven't given up on that."
Mr Johnson dismissed his rival's pledge to cancel student debts for some entrepreneurs, saying: "I think people, a lot of people, would automatically be defining themselves as entrepreneurs."
"I think the more sensible things to look at are the interest rate, and a reduction of the interest rate, also looking at the cost of maintenance because I think those are very, very high and that people are paying a lot of money back over a long time."
The candidates are set to face each other at an ITV debate on 9 July and at an event hosted by the Sun newspaper and talkRADIO on 15 July.