Video report by ITV News Political Editor Robert Peston
Prime Minister Johnson says he wants a Canada-style trade deal with the EU, but the EU is demanding the UK sign up to regulations on state subsidies, environmental standards and workers' rights.
The EU wants a "level playing field" and to ensure "competition remains open and fair" before such an agreement can be signed.
The EU's chief Brexit negotiator Mr Barnier also wants the EU to have access to Britain's fishing waters.
He says the UK must agree to these demands if it is to win a deal like Canada's, which would remove tariffs on almost all imports and exports of goods.
But Mr Johnson, speaking in Greenwich, said there's "no need" for Britain to sign up to the "full panoply" of Brussels' regulations and insisted British fishing grounds are "first and foremost" for UK boats.
He told business leaders in London he'd rather fall back on an Australia-style arrangement when the 11-month transition period ends, rather than align with the EU's demands.
The problem with Mr Johnson's plan B is that Australia doesn't have a free trade agreement with the EU and critics have branded it "no-deal in all but name".
Mr Johnson said: "The question is whether we agree a trading relationship with the EU comparable to Canada's or more like Australia's and I have no doubt that in either case the UK will prosper mightily."
He says he won't comply with demands on "competition policies, subsidies, social protection, the environment or anything similar" because the UK left the EU to escape such regulations.
The UK should not be made to accept rules from the EU, Mr Johnson said, "any more than the EU should be obliged to accept UK rules".
After the speech Downing Street said the UK will agree to some alignment with the EU as long as it is not overseen by the European Court of Justice (ECJ).
A spokesman said there is no need to go beyond the provisions set out in the Comprehensive Economic and Trade Agreement between Canada and the EU.
Boris Johnson explains to Robert Peston why he will not accept EU rules:
"The key point is nothing that involves ECJ oversight and we don't believe that there's any need to go further than any of the provisions the EU has agreed with Canada," he said.
Foreign Secretary Dominic Raab said the government's post-Brexit Global Britain Strategy will be based on "three pillars" of being a good ally, a defence of free trade and doing good in the world.
In a Commons statement, Mr Raab said the first pillar of the strategy is "to continue to prove that we are the best possible allies, partners and friends with our European neighbours".
The next pillar is for Britain to be an "energetic champion of free and open trade, to boost small business, to cut the cost of living, to create the well-paid jobs of the future".
The final pillar aims to see the UK as "an even stronger force for good in the world", including a focus on human rights and climate change.
The Foreign Secretary also said he will visit Australia, Japan, Singapore and Malaysia over the next fortnight in order to discuss future trade arrangements.
Another trade deal expected to be discussed is one with the United States.
And despite a presidential election taking place there later this year, the US ambassador to the UK believes there is time for a deal to be reached before President Donald Trump faces re-election.
Asked by ITV News Political Editor Robert Peston whether the US could "focus" on a trade deal before the November 3 election, Woody Johnson said: "Absolutely.
"The president has made that point many many times about how important this deal is to him and the US," he added.
"I think you can get a deal if you have willing partners on both sides and you're willing to get the emotions down and use your best efforts, a deal can be done."
Mr Barnier warned that even if the UK and EU were able to agree a "best-in-class" trade deal, it would still not be "business as usual".
He said rules of origin and customs formalities would apply between the UK and the EU and access to EU markets would be subject to "certification and market authorisation and supervision activities".
Mr Barnier said the more the UK was prepared to maintain common standards with the EU, the higher quality access it would get to EU markets.
"This will be up to the UK to decide. Will it continue to adhere to Europe's societal and regulatory model in the future or will it seek to diverge?" he said.
Mr Barnier said the EU would be "clear headed" in its negotiations with the UK.
Brussels' chief negotiator said: "Our aim is to conclude an ambitious partnership with the United Kingdom.
"But we will remain clear headed. The most ambitious partnership is the one that we had, because we were in the same union."
He added: "When you are not a member of the European Union then, objectively speaking, your position is different and less favourable."
Meanwhile, Labour leader Jeremy Corbyn said any claim by the prime minister that a no-deal Brexit had been avoided, should be taken "with a massive pinch of salt".
The UK, upon its departure from the bloc last week, entered an 11-month transition period in which it has no representation in Brussels but continues to follow EU rules while it works to agree a new relationship with the remaining 27 member states.
It might be March before formal trade talks between the two sides commence, as the EU27 need to agree a joint position.