A cabinet minister said leader of the pro-Brexit European Reform Group Jacob Rees-Mogg also supports the plan, while other Brexiteers said it could be worth exploring in more detail.
"Britain stands on the brink of the greatest crisis since the Second World War", said Johnson, a former Financial Times journalist who voted to stay in the European Union in the 2016 referendum.
At the party conference, Labour delegates left open the possibility of backing a second referendum if it could not force a general election.
A cabinet minister said the plan has been developed to ensure the worst elements of a no-deal Brexit could be eased, but they also pointed out it could be beneficial to the United Kingdom in the long run.
"Given that the reality of Brexit has turned out to be so far from what was once promised, the democratic thing to do is to give the public the final say", Johnson said.
Such a choice was a "failure of British statecraft on a scale unseen since the Suez crisis", he added.
Asked if his brother Boris Johnson had lied to voters during the referendum about Brexit, the former transport minister said: "In the campaign there were undoubtedly promises made that have shown to be undeliverable".
The choice being presented to the British people was no choice at all, he said in an online article, announcing his resignation. "If these negotiations have achieved little else, they have at least united us in fraternal dismay".
Jacob Rees-Mogg, a leading Brexiteer in May's party, told BBC Radio he rejected the call for another referendum but agreed with the criticism of the deal.
Britain wants to strike an agreement with the EU as quickly as possible and it would be desirable for Britain and the EU to reach a deal at a special EU summit in November, he added.
"Boundless admiration as ever for my brother Jo", he said.
"There are many Conservative MPs who share Jo Johnson's serious concerns".
And his sharpness with words was evident in his resignation statement as he insisted "Britain stands on the brink of the greatest crisis since the Second World War". "It does not remotely correspond to the mandate of the people in June 2016".
Jenny Chapman, Shadow Brexit Minister, said: "Jo Johnson is the eighteenth minister to resign from Theresa May's government".
But calls for a second Brexit referendum have been attacked by ex-first secretary of state Damian Green.