It comes as the Tories brace for a leadership contest, with the prime minister expected to outline her timetable for departing Downing Street if a key piece of Brexit legislation is voted on by MPs in the first week of June.
But an analysis for The Independent by BMG pollsters before the prime minister met with the executive of the 1922 Committee shows the public at large are vastly apathetic towards the likely contenders.
Over half of those polled - 55 per cent - said they did not know, or replied: "None, haven't heard of'.
Asked who would make the best Conservative prime minister if Ms May was to stand aside as prime minister, Mr Johnson, who formally declared his candidacy this week, received the most support - on 11 per cent.
His closest contender - the hardline Brexiteer Jacob Rees-Mogg - attracted six per cent of those polled, while Philip Hammond, the chancellor, Ruth Davidson, the Scottish Tory leader, and home secretary Sajid Javid all received four per cent.
On three per cent are the work and pensions secretary, Amber Rudd, and the environment secretary Michael Gove. Ex-Brexit secretary Dominic Raab, who is widely expected to enter the race, received the backing of just one per cent of respondents.
Simon Wood, the head of insight at BMG, said that while Mr Johns leads the field, "his support is dwarfed by the scale of rejection the public are showing towards leading Conservatives".
He continued: “Over half the population are unable to nominate anyone they would want. The Conservatives are in real danger of choosing a leader rejected by the public and setting themselves on the path to electoral disaster”
“Boris Johnson may be the lead candidate but the public apathy to leading Conservatives simply makes him the least worst option”.
It comes as Mr Johnson emerged as the early favourite to succeed the prime minister among the Conservative Party grassroots on Friday.
In a separate YouGov poll for The Times of party members, who will have the final say in the contest, rather than the general public, put the former mayor of London on 39 per cent - three times the 13 per cent for Mr Raab.
Speaking earlier this week, Mr Johnson confirmed his intention to stand for the leadership at a private event in Manchester, declaring that he had "boundless appetite to try to get it right".
He continued: "I'm going to go for it. Of course I'm going to go for it. I don't think that is any particular secret to anybody. But you know there is no vacancy at present."
Source note: BMG Research interviewed a representative sample of 1,541 GB adults online between 7 and 10 May. Data are weighted. BMG are members of the British Polling Council and abide by their rules.