The prediction, based on a poll of 6,124 adults over the last two days by YouGov for The Sun, is that Nigel Farage's Ukip will gain most votes with 27%, despite him being involved in a number of racist and homophobic controversies in the lead up to election day.
Labour is projected to come second with 26%, the Tories third with 22% of the vote, followed by the Greens on 10% and the Lib Dems with 9%.
The poll suggests that Ukip and Labour will both have 22 MEPs, an increase of nine each on the last elections in 2009. The Conservatives are expected to get 16 of the 70 European Parliament seats being voted on in England, Scotland and Wales, 10 fewer than before, while the Greens will get four - an increase of two.
The Lib Dems are expected to lose eight, leaving them with just three MEPs, while the far-right BNP will lose both its MEPs, the poll predicts.
Farage shrugged off allegations that Ukip was a racist party as "rot" as he revealed he was "reasonably" confident about its election chances.
Before casting his ballots at Cudham C of E Primary School in Westerham, Kent, the Ukip leader told reporters: "If we get what we like things will never be quite the same again."
He added: "Waking up this morning I thought, 'What on earth do I do?'. I have been campaigning for months and I feel a bit vacant now."
Asked about claims of racism in his party, he said: "I think the allegations of the political establishment are against us.
The poll also found that less than a quarter of voters planned to go to the polling booths to actually influence the European Parliament, with 63% saying they would do so to express their views on the political scene in Britain and its relationship with the EU.
Immigration and membership of the EU were the key issues for voters, but 42% still said they preferred to stay in Europe.
Meanwhile, a leaked internal Liberal Democrat document suggests the party is braced for the possibility of failing to win any seats at all in the European parliamentary elections, the Guardian claimed.
The document, the contents of which the newspaper says it has seen, is said to advise senior party figures to say it was "expected" should it win no seats.
If the party wins between zero and two MEPs, Lib Dem spokesmen have been briefed to say they are "disappointed with the result but the party remains resolute and this was expected at this point in an electoral cycle".
Two to three MEPs would be "a good result considering the circumstances", while three to five MEPs would be a "very encouraging result and is much better than almost everyone predicted", the Guardian said.
The BBC Charter includes and obligation to "ensure that controversial subjects are treated with due accuracy and impartiality."
It also says the Lib Dem's best hopes are to win five seats, less than half of its 11 MEPs from 2009. The newspaper said the document was presented to the Wheelhouse group, the body chaired by general election co-ordinator and former Lib Dem leader Paddy Ashdown.
A Lib Dem spokesman told the Guardian: "It's no surprise that, like all parties, we prepare for all sorts of outcomes. We know it will be a tough night but we are proud of our campaign.
"Nick Clegg is the only leader that has stood up to Ukip's divisive politics and defended British jobs and Britain's place in Europe."
Watch Mehdi's Minute: Ukip and European Elections Special
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