Senior Cabinet ministers have insisted the UK is prepared to walk away from Brexit talks without a deal, on the second anniversary of the referendum.
Liam Fox said Theresa May was "not bluffing" over her threat to quit negotiations, while Boris Johnson called for a "full British Brexit".
It comes as anti-Brexit campaigners, who want the public to have the final say on the UK's departure, prepare to march in London later.
They say Brexit is "not a done deal".
People's Vote - which wants a referendum on any exit deal - said people must make their "voices heard" about the "damage" of leaving next year without agreement.
Speakers at the demo will include actor Sir Tony Robinson and campaigner Gina Miller, who fought a successful legal battle last year to ensure the UK could not trigger talks on leaving without the approval of Parliament.
The UK voted to leave the EU by a margin of 51.9% to 48.1% in a referendum held on 23 June 2016.
The UK is due to leave on 29 March 2019, 46 years after it first joined the European Economic Community, the forerunner to the EU.
But the People's Vote campaign says this should happen only if the withdrawal deal negotiated by Mrs May and the other 27 EU members is approved in another public vote.
International Trade Secretary Liam Fox told the BBC it was in the interests of both sides to have a deal - but it was "essential" the EU understood that the UK could walk away if the terms offered were not good enough.
"The prime minister has always said no deal is better than a bad deal," Mr Fox said in an interview with the BBC's political editor Laura Kuenssberg, which was recorded on Wednesday - before Friday's warning from Airbus that it might cease manufacturing in the UK in such a scenario.
"It is essential as we enter the next phase of the negotiations that the EU understands that and believes it... I think our negotiating partners would not be wise if they thought our PM was bluffing."
'Bog roll Brexit'
Meanwhile, Brexit Secretary David Davis told the Daily Express the prime minister was going to get a "good deal" from Brussels and Brexit was going to be "fantastic".
"The best option is leaving with a good deal but you've got to be able to walk away from the table," he said.
And writing in the Sun, Foreign Secretary Boris Johnson warned the prime minister not to allow "bog roll Brexit" that is "soft, yielding and seemingly infinitely long" - calling for a "full British Brexit" instead.
Mr Johnson said people "just want us to get on with it".
By BBC political correspondent Nick Eardley
Two years on from the referendum, there are two very different messages today.
One is that Brexit is not a "done deal".
That's the argument from the Liberal Democrat leader Sir Vince Cable who will take part in a march in central London today.
On the other side, there is optimism and defiance from key Brexiteers like Boris Johnson.
Mr Johnson says the UK is confident and open.
The government firmly opposes a vote on the final deal - Mr Johnson believes people want the government to just get on with it.
Nine months before the UK is due to officially leave the EU, there are still very different visions.
Labour said Mr Fox's comments about a no-deal Brexit were the "height of irresponsibility".
"The next time Liam Fox parrots the slogan no deal is better than a bad deal he should give some thought to the 14,000 people who work for Airbus, and the thousands of other people who have jobs dependent on trade with Europe," said shadow Brexit minister Jenny Chapman.
Both the prime minister and Labour leader have rejected calls for another public vote, saying the will of the people expressed in the 2016 ballot was clear, although many Labour MPs now want another referendum.
Organisers of Saturday's demo say people "from all walks of life" will be present, demonstrating the "growing popular demand" for another vote.
Beginning in Pall Mall and ending outside the Houses of Parliament, the protest is part of a "summer of action" by campaign groups designed to increase pressure on Theresa May and Jeremy Corbyn.
By taking the UK out of the EU's single market and customs union, they say the Conservative government "remains intent" on a so-called hard Brexit that will - they say - destroy jobs and damage public services.
The leader of the Liberal Democrats, Sir Vince Cable, who will be at the march, told BBC Breakfast: "I think the public in general... do see there is a mess."
He added: "We've only got a year to go. And I think for the big companies that employ hundreds of thousands of workers in the UK... they want some clarity about what the trading relationships will be and there is absolutely none whatever."
But Conservative MP Peter Bone - who supports Brexit - said if there were a second vote, the leave campaign would win again.
"The vast, vast majority of people, whether they are Leavers or Remainers, just want us to get on and come out this dreadful European Union super-state," he said.
"There were 17.4 million people that voted for leave and if there are a few thousand in London complaining about it - that doesn't seem to really make much difference."
The government is giving Parliament a vote on the final deal, if one is reached, in the autumn - but it remains unclear what will happen if they reject it.