The survey, carried out by YouGov for The Times, has the Tories on just 10%, behind the Greens on 11%.
The Liberal Democrats are in third place on 15%, closely behind Labour on 16%, but the big frontrunners remain the Brexit Party, with 34%.
Nigel Farage’s new party have increased their share from 30% last week, as it continues to be a strong favourite in the European elections.
Labour has dropped 5% from 21% the previous week.
The newly formed Change UK has also slumped, from 9% last week down to 5%. Ukip brings up the rear on 3%.
YouGov surveyed 2,212 adults for its survey.
However, separate YouGov results showed that voters believe the Brexit Party and Change UK will "eventually fade" from British politics within a decade.
A YouGov survey revealed 63% of people thought the Brexit Party would "probably not be a force in British politics in 10 years", compared to 13% who thought it was here to stay.
Approximately 56% thought Change UK would disappear over the next decade, with just one in 10 believing it will "likely remain an important part of British politics".
Some 70% thought Labour would continue to exist in 10 years, with 71% believing the Tories would survive the next decade, and just under half, 45%, thinking the Lib Dems were here to stay.
The poll found the Green Party was seen as the third most likely to still be around in 2029.
YouGov political research manager Chris Curtis said: "The upcoming European Parliament elections will be the first outing for two brand new parties: Change UK - The Independent Group and Nigel Farage's Brexit Party.
"The data shows the public aren't yet convinced that these newer forces will become a permanent feature in British politics, although both these parties will be hoping to prove the public wrong, and a strong showing in next week's election could give them momentum and help them stand out in an increasingly crowded field."
YouGov polled 1,867 adults between April 30 and May 1.
The new findings came as a general election poll found the Brexit Party had overtaken the Tories for the first time.
Mr Farage's party was one point ahead of the Conservatives in what would be the Tories' worst ever result, according to the ComRes survey of voting intentions.
That level of support would see the Brexit Party win 49 seats, becoming the UK's second biggest party after Labour, with 137.
Andrew Hawkins, chairman of ComRes, described the findings as a "disaster", adding: "If the Conservative leadership contenders are not careful, there will be no party for them to lead."