Anti-Brexit campaigners who want the public to have the final say on the UK’s departure will take to the streets later to argue it is “not a done deal”.
The London march comes on the two year anniversary of the 2016 vote to leave.
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.
But Brexiteer cabinet minister Liam Fox said Theresa May was “not bluffing” when she said this could happen.
The international trade secretary 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 would walk away if the terms offered were not good enough.
Meanwhile 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.
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 UK voted to leave the EU by a margin of 51.9% to 48.1% in a referendum held on 23 June 2016.
As it stands, 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 only happen if the withdrawal deal negotiated by Mrs May and the other 27 EU members is approved in another public vote.
Saturday’s demo, which will begin in Pall Mall and culminate outside the Houses of Parliament, is part of a “summer of action” by campaign groups designed to increase pressure on Theresa May and Jeremy Corbyn.
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 is clear, although many Labour MPs now want another referendum.
Speakers at the event, which organisers say is expected to be the biggest anti-Brexit march to date, will include actor Sir Tony Robinson and 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.
Organisers say people “from all walks of life” will be present, including farmers, doctors, teachers, delivery drivers, students, fishermen and veterans, demonstrating the “growing popular demand” for another vote.
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 destroy jobs and damage public services.
Labour MP Chuka Umunna, who will join Lib Dem leader Sir Vince Cable and Green Party leader Caroline Lucas among those who will address the crowds, said momentum was building behind calls for the public to decide on the final the terms of exit.
“With the negotiations in chaos, the government making a total mess of Brexit and with Parliament paralysed, now is the time for people to take back control of the process,” said Mr Umunna.
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.
Ministers insist the prospect of leaving the EU without any agreement remains firmly on the table.
“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.”
Urging the EU to show more flexibility in talks over its future relations with the UK, Mr Fox said the failure to reach a mutually satisfactory deal could have a “severe” impact on countries like the Republic of Ireland and the Netherlands, key markets for British trade.
“That cannot be what the EU 27 really want to see,” he said. “This ultimately has to be about a people’s Brexit, not a bureaucrats’ Brexit.”
Reflecting on events since the referendum, he said warnings during the 2016 campaign of “economic Armageddon” if people voted to leave had been unfounded and the UK had been given a “very big vote of confidence” since then by foreign investors.
But 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.
In his interview in the Sun, Mr Johnson said people “just want us to get on with it”.
“They don’t want a half-hearted Brexit. They don’t want some sort of hopeless compromise, some perpetual push me-pull you arrangement in which we stay half-in and half-out in a political no man’s land,” he said.
Meanwhile, Mr Davis said Brexit is a “big national project, there is nothing more important”.
He added that Britain is at an advantage in world trade because English is “the best language in the world for doing commerce, science and medicine and so on”.