The Ipsos MORI survey for the Standard showed the Conservatives on an unprecedented low, in fourth or even fifth position and struggling to beat the Greens.
The dismal score, less than half the 20 per cent of the Liberal Democrats, would be the worst Tory result ever in the European Parliament direct elections since they began 40 years ago.
It would also be the lowest support for the Conservatives in a national election since at least the First World War.
Nigel Farage’s Brexit Party is on course to be the winner, with the poll putting it on 35 per cent.
The Brexit Party’s backers also say they are most likely to vote, with more than half its support coming from people who voted Tory in the 2017 general election.
Labour is in third place on 15 per cent, with voters “confused” over the party’s Brexit position as Jeremy Corbyn has sat on the fence over a second referendum.
Even some Labour MPs in Leave seats now accept it would have been better to have gone into the election backing another public vote so they would have a clear position to try to sell.
The Greens are on 10 per cent, with Change UK and Ukip both on three per cent. But the febrile political atmosphere means the results could be volatile.
What is almost certain though, is that when the results are announced on Sunday night it will be another Tory rout, with the party’s vote share set to be far worse than the 2017 election when Mrs May lost the Tory Commons majority.
Gideon Skinner, head of political research at Ipsos MORI, said: “Both Labour and Conservatives are suffering in this election, while the Brexit Party looks set to be the winner.
“We’ve seen how Conservative Party voters in particular dislike their party’s approach to Brexit, and prefer Nigel Farage’s party line.
“Labour supporters are confused over exactly what Labour would do about Brexit if it were in power.”
However, Mr Skinner also stressed that a low turnout could impact on the results, that there are still a “high number” of voters who may change their mind and the emergence of new parties is adding to the “volatility”.
One in eight potential voters who supported the Conservatives in 2017 are now intending to switch to the Lib Dems, with just a fifth sticking with Mrs May’s party.
The poll also showed Labour losing support to the Lib Dems, with 22 per cent of its 2017 voters deserting it for the party, and 15 per cent to both the Brexit Party and the Greens.
Some 90 per cent of Brexit Party voters say they have definitely decided to vote for it.
In contrast, 46 per cent of Green voters, 44 per cent of Labour backers and 40 per cent of Lib Dem supporters say they might change their mind before voting, highlighting the array of options for people who oppose Brexit.
Just over half of people planning to vote Conservative say they may not do so in the end.
Two thirds of voters say they have definitely decided who to back, low by historical standards, and just 61 per cent of Britons say it is important to them who wins the European elections, compared with 86 per cent in the last three national polls.