The survey by the pollsters YouGov follows the prime minister reiterating in the Commons he would under no circumstances request a Brexit delay beyond 31 October.
The remarks came as legislation forcing the government to request an extension from Brussels until 2020, if a deal is in play by 19 October, received royal assent.
According to the fresh poll, when respondents were asked what Mr Johnson’s next step should be, 28 per cent said he should break the law by refusing to ask the EU for a Brexit deadline extension.
Half of those polled (50 per cent) said the prime minister should not break the law while a further 22 per cent replied “don’t know”.
But among those who voted Conservative at the 2017 general election, 5o per cent said that Mr Johnson should break the law, compared with 34 per cent who said he should adhere to the extension request.
Of those who voted Leave at the EU referendum in 2016, 52 per cent believed the government should flout the law while 28 per cent disagreed.
Speaking in the Commons ahead of the suspension of parliament and MPs blocking his bid for a general election a second time, the prime minister said: “I will not ask for another delay.”
He continued: “The public have had enough of the delectable disputations in this House and I must warn Hon Members that their conduct has gravely undermined this House in the country.”
But Jeremy Corbyn said the prime minister had given a “clear indication” that he does not intend to follow the law passed by MPs that requires him to ask for an extension mid-October.
He also asked Mr Johnson to respect the rule of law, adding: “This parliament is not a platform for the prime minister’s games. It is a chamber in which the people’s representatives hold the executive to account.”
Ian Blackford, the SNP’s leader at Westminster, also lashed out at the remarks in the chamber, claiming: “The prime minister is saying with those words that he’s going to ignore an act of parliament. That he is going to ignore the law. I simply say to the prime minister: be careful.”
“You occupy the highest office in the land and what you’re demonstrating to the people of the UK is that the law doesn’t matter. That’s a very serious situation to be in.”