That level of support would see the Brexit Party win 49 seats, becoming the UK’s second biggest party after Labour, with 137, the Press Association reports.
Andrew Hawkins, the chairman of ComRes, described the poll as a “disaster”, adding: “If the Conservative leadership contenders are not careful, there will be no party for them to lead.”
The news in the Sunday Telegraph follows the calamitous council elections, where May oversaw the loss of nearly 1,300 Tory councillors, and comes ahead of a predicted wipeout in the European elections in the next fortnight.
The poll shows the Conservatives would lose 46 seats to the Brexit Party, dethroning Defence Secretary Penny Mordaunt, Health Secretary Matt Hancock and party chairman Brandon Lewis.
And Labour would take the scalps of Boris Johnson, Iain Duncan Smith and Sir Graham Brady, the chairman of the 1922 committee, with the Tories retaining support from less than half of those who voted for them in 2017.
Jeremy Corbyn would be able to lead a minority government with 27% support, leaving the Brexit Party with 20% and the Conservatives 19% support, according to the poll commissioned by Brexit Express.
Brexit Express is a campaign group run by Jeremy Hosking, a major Tory donor who has now given £200,000 to Farage’s party.
The ComRes poll chimes with another recent poll by Opinium, which showed the Brexit Party snapping at the Tories’ heels in a Westminster election, where they would be just one point behind.
According to Opinium, Labour would be out in front with 28% support, followed by the Tories on 22% and the Brexit Party on 21%.
Adam Drummond, head of political polling at Opinium said it was “remarkable” the new party could be on the cusp of overtaking the party of government.
Although Drummond cautioned the European election in less than two weeks could be “bleeding into” and “inflating” the Westminster results, the level of support was still notable.
He said: “It would be reasonable to assume that this would fall back in an actual general election campaign, but the fact a party that is less than six months old is now vying with the governing party for second place is remarkable.
“The Tories’ reliance on Leave voters seemed sensible in the aftermath of the referendum.
“But raising expectations of the kind of deal the UK could get and using the phrase ‘no deal is better than a bad deal’ so relentlessly was always going to open the party up to this kind of challenge once those expectations could not be met.”
The two main parties, Labour and the Conservatives, continue to drop votes, with Labour falling five points and the Tories four in the last fortnight.
But the Brexit Party and the Liberal Democrats are reaping the rewards of public frustration and are both on the up, with the former rising five points and the latter four – taking the Lib Dems to fourth place with 11 points.
When it comes to the European elections, the Brexit Party is galloping ahead into first place with 34% support – doubling the existing gap to 13 points.
Labour, coming second with 21%, has fallen seven points in the last fortnight and the Lib Dems are in third position with 12%, having risen five points.
The Tories have dropped another three points to stagger into fourth place with 11% support.
Drummond said Remainers were split on which party to support and 57% would like to see a pro-Remain alliance.
He said: “On the European elections, while the question of which party Brexit voters should back was settled some time ago, the equivalent for Remain voters is still ambiguous.
“The picture is most stark when we split out Leave and Remain voters – while 63% of Leavers say they will vote for the Brexit Party in the European elections, the most popular party among Remainers (still Labour) only has 31% versus 22% for the Lib Dems and 14% for the Greens.
“Interestingly, Brexit voters have deserted the Tories to such an extent that the Conservatives actually have a higher share of the European vote among Remainers (12%) than among Leavers (11%).”
Opinium Research carried out an online survey of 2,004 UK adults over 18 years old between May 8 and 10 and results have been weighted.