The In camp are ahead of Leave by 52% to 48%, according to an Ipsos Mori survey for the Evening Standard.
To add further uncertainty to the outcome, 12% of those polled said they could switch sides as they head for the polling booth.
With turnout key to the result, torrential rain storms in Remain stronghold London meant the unseasonally bad weather could deter voters casting their ballots.
Ukip leader Nigel Farage insisted his side could pull off an historic victory if “soft” In voters stayed home.
“Actually I do think we are in with a very strong chance, I do genuinely. But it’s all about turnout and those soft Remainers staying at home. I do think that the people who have decided to vote Leave have a real conviction and passion,” Mr Farage said.
However, early indications showed long queues snaking outside voting centres as the nation braved the storms to decide Britain’s future role in Europe.
But some polling stations were forced to close, and many others are flooded due to the heavy rain sweeping the capital.
Kingston upon Thames Council in south west London moved two polling stations after they became inundated with water.
Prime Minister David Cameron ignored questions about the weather, saying only “Good morning” to the gathered media from across the world, as he and wife Samantha cast their votes at Methodist Hall in Westminster.
Labour leader Jeremy Corbyn arrived in a positive mood to cast his vote in his Islington constituency in north London.
Asked if he was feeing confident, Mr Corbyn smiled and said: “Extremely, it’s a very good day.”
On the outcome of the referendum, he joked: “You could either check the wind or check the bookies,” adding “the bookies usually get it right”.
Pro-Leave Justice Secretary Michael Gove said he was feeling “quite excited” as he accompanied his wife, Sarah Vine, to a polling station in North Kensington.
The referendum has seen one of the longest, and most personally bitter, campaigns in recent British political memory.
Both sides of the campaign have been locked in fierce fighting for months, and things came to a frenetic close on Wednesday as senior politicians criss-crossed across the country to try to sway undecided voters.
The Prime Minister and his Remain colleagues from across the political spectrum have warned of the potentially severe economic consequences of a Brexit vote amid fears of financial market turmoil and another recession.
But Leave campaigners, led by Tory heavyweight Boris Johnson, have urged voters to “take back control” of the country.
Mr Farage would not elaborate on his reasons for missing Wednesday night’s final Channel 4 debate of the campaign, saying only that it was “for family reasons”.
He came in for particularly stern criticism after unveiling a Brexit poster showing a queue of hundreds of immigrants arriving in Europe with the slogan “breaking point”.
And Leave campaigners were left furious after Chancellor George Osborne made use of Treasury research to warn that quitting the EU would result in households being £4,300 a year worse off.
Meanwhile, dozens of celebrities have intervened during the course of the campaign to make their feelings known.
Footballer David Beckham, James Bond actor Daniel Craig and Harry Potter author JK Rowling were just three of the high-profile names to back the Remain campaign, while Leave won support from the likes of comedian John Cleese, former cricketer Sir Ian Botham and former England football player Sol Campbell.
A record number of voters are eligible to take part in the referendum, with the Electoral Commission putting the number at 46,499,537.
Polling stations will close at 10pm.
Fieldwork for Ipsos Mori’s survey of 1,592 people was carried out until 9pm on Wednesday night.
Concern over Turkey’s future relationship with the EU dominated national EU referendum coverage on the final day of campaigning, while Ed Miliband told voters in Leeds that there is no “magic bullet” to Britain’s immigration crisis.
Former Labour leader and Doncaster North MP Ed Miliband and former shadow work and pensions secretary Rachel Reeves MP were at Mitie, a business strategy company in Thorpe Park yesterday morning where they spoke to staff.
Mr Miliband, who is backing Labour In For Britain, said one solution to immigration levels is more investment in public services to relieve pressure.
Ms Reeves has said that the concerns raised during the referendum should be a wake-up call to all political parties and the issue of immigration from the EU needs to be looked at again.
However, during the final day of campaigning, David Cameron faced yet more criticism that he had not been straight with the British public over Turkish membership of the EU.
Last night Turkish President Erdogan’s Chief Adviser, İlnur Çevik, said that the Turkish government was ‘flabbergasted’ that their ‘chief supporter’ was saying something different in public to what he has told them in private.
Leeds Central MP and shadow foreign secretary Hilary Benn addressed crowds in Sheffield and Vote Leave campaigner Boris Johnson arrived by helicopter.
Using his well-oiled rhetoric of national pride, Mr Johnson told activists: “I do think that we are on the verge, possibly, of an extraordinary event in the history of our country and indeed in the whole of Europe.”
However Vote Leave have had a bumpby start to the day when two Conservative MPs wrote to chief executive Matthew Elliott asking him if the organisation will return a £600,000 donation given to them by a former member of the British National Party.
Justice Secretary Michael Gove said he was not aware that Gladys Bramall, who donated the money to Vote Leave, featured on a leaked BNP membership list from 2006.
But Mr Gove insisted on the radio this morning that if she held the “objectionable” anti-immigrant views of the BNP then Vote Leave would return it and not use it in campaigning.