The Liberal Democrats have leapfrogged Labour sending Jeremy Corbyn's party into third place, according to one poll.
But despite a surge, reported by YouGov, Vince Cable's party is still a distant second to Nigel Farage's Brexit Party.
The Conservatives could come fifth in the elections on May 23 - with the poll suggesting they might win just 9% of the vote.
The YouGov study put Sir Vince Cable's party, which is campaigning with a Stop Brexit message, on 16%, one point ahead of Labour , while the Tories dropped to single-digit levels of support on 9%.
The Brexit Party was on 35% according to the survey of more than 7,000 people.
The YouGov study again showed the Tories in fifth place, behind the Greens on 10%. Change UK were on 5% and Ukip on 3%.
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YouGov's Adam McDonnell said: "Both the Conservatives and Labour look set to lose a majority of those who voted for them in the 2017 general election.
"Only one in five (20%) of those who voted Conservative just under two years ago are sticking with the party for next week's European Parliament vote, with 62% instead planning to back the Brexit Party.
"Labour is also losing a large proportion of their voters, mainly to Remain parties (Lib Dem 21%, Green 16%, Change UK 6%)."
Compared with YouGov's previous study for The Times on May 8-9, the Lib Dems and Brexit Party were up one point, Labour and the Conservatives both down one.
YouGov surveyed 7,192 British adults between May 12 and 16.
