In response, Mr Johnson said it would be similar to Switzerland's relationship with the European Union.
A 30-year schism inside her party over Europe helped sink the premierships of the past three Conservative prime ministers - Mrs Margaret Thatcher, Mr John Major and Mr David Cameron.
Speaking to reporters on her way to NY to attend the United Nations General Assembly, May said she welcomed a subsequent acknowledgement from European Council President Donald Tusk that the bloc still wanted to strike a deal.More news: Hurricane Rosa Expected To Soak Los Angeles, Increase California Surf
The Cabinet minister, who was handed the Foreign Office brief in July after Boris Johnson sensationally quit, also backed the Prime Minister's resolve and warned the European Union and doubters in the United Kingdom that "underestimating Theresa May is one of the biggest mistakes that you could make right now".
Asked if Mr Barnier gave any indication that Labour's plan would be easier to implement than the Chequers plan, Mr Corbyn said: "He didn't offer and we didn't ask him for opinions on that because that would be going over the line".
"Don't underestimate Theresa May, I think what's happening at the moment is that after that very hard general election, maybe some people in Britain didn't appreciate the steal at the heart of that lady".More news: The Walking Dead Spoiler: Lauren Cohan Might Return in Season 10
"There has been a collective failure of government, and a collapse of will by the British establishment, to deliver on the mandate of the people".
He advocates a looser "Canada-style" agreement that would leave Britain freer to strike new worldwide trade deals. He said that in such a situation voters would not be choosing between Labour and Conservative but would be voting to establish a strong government.
"We need people to look at the very practical proposals that the Prime Minister has put on the table and I think when they look at them they realise that actually this is a very sensible way to deal with all the different tensions, all the different trade-offs". "At the moment, I cannot see how I can support Chequers".More news: Judge: Democrats in Congress can sue Trump over emoluments
"If Labour does the right thing for the country, there's a good chance that we can hold a people's vote and stop Brexit", he said.
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