The former foreign secretary, already odds-on favourite with bookmakers, has a massive lead over his nearest rival, fellow Brexiteer Dominic Raab.
Mr Johnson is the first choice of almost one-in-four party members, according to a YouGov poll for The Times conducted this week.
Cabinet ministers Jeremy Hunt, Michael Gove, Sajid Javid, Andrea Leadsom, Matt Hancock and Rory Stewart are all trailing way behind on single figures.
The poll is hugely significant because it is Conservative Party members who will elect Mrs May’s successor in a ballot of the top two candidates chosen by Tory MPs.
It suggests that unless his leadership bid is blocked by a stop Boris sabotage by MPs, he is on course to storm to victory by a big margin in the ballot of party members.
Mr Johnson is first choice to replace Mrs May among 39% of the Tory membership, according to YouGov, with Mr Raab on just 13%.
In a head-to-head run-off, Mr Johnson would decisively beat Mr Raab by 59% to 41% and trounce every other candidate by an even bigger margin.
But the poll confirms he is also a divisive figure among Remain-supporting Tories, coming fifth among those who voted Remain in the 2016 referendum, behind Mr Javid, Mr Gove, Mr Hunt and Mr Stewart.
Some 31% of Tory members think he would be a “poor” leader, including 65% of those who voted Remain, and he is less popular in London than the rest of the country despite eight years as London mayor.
Mr Gove and Mr Javid are the first choice of only 9% of Tory members, Mr Hunt 8%, Andrea Leadsom and Penny Mordaunt 5%, Rory Stewart 4% and Matt Hancock just 1%.
Mr Johnson’s front-runner status means he will now come under sustained attack from opponents in other political parties. SNP leader Nicola Sturgeon led the way, branding him a “charlatan”.
She said at the SNP’s Euro-elections manifesto launch in Glasgow: “Boris Johnson, the guy who misled people in the Brexit vote, the guy who has only ever put his own interests first as far as I can see – a complete and utter charlatan in my opinion.
The prospect of him becoming prime minister of the United Kingdom is, I think, one that will horrify many people across Scotland.
Among Tory members, however, Mr Johnson is seen as the most likely figure to win a general election if he became leader, with 70% saying he would be victorious.
Some 42% think Mr Raab would win, 39% Mr Javid, 32% Mr Gove and 29% Mr Hunt. Mr Johnson claims he is the only candidate capable of taking on Nigel Farage.
The poll also suggests two thirds of Conservative Party members back a no-deal Brexit, with 66% backing it as their first option, 13% wanting a return to Brussels to get a deal which could attract remaining Tory and DUP MPs, 12% backing the existing deal and only 6% supporting a deal with Labour.
More than eight out of 10 Tory members oppose a second referendum, even if no-deal is one of the options on the ballot paper.
Like many recent polls, this one makes gloomy reading for Mrs May, with 64% of Tory members opposing her Brexit deal, and 79% saying she is doing “badly” as PM and should stand down.