A YouGov poll released at 10pm suggested Remain was at 52% and Leave on 48% in the Brexit referendum.
Sky News reported Ukip's Nigel Farage as saying it "looks like Remain will edge it".
A "Brexit Party" organised by the Leave.EU campaign group kicked off at Millbank Tower in central London just before polls closed with a performance by British soul singer Kenny Thomas who found success in the 1990s.
Mr Thomas was followed on stage by Gwen Dickey from the American group Rose Royce - perhaps best known for the hit single Car Wash.
However, polls suggesting the result remains too close to call.
UKIP leader Nigel Farage says it "looks like 'Remain' will edge it" #EUref
— Sky News Newsdesk (@SkyNewsBreak) June 23, 2016
Prime Minister David Cameron faces a tense night as votes are counted across the UK, with a Leave vote widely expected to spell the end of his premiership.
The final poll released just hours before voting stations closed their doors put the In side ahead by a margin of 52% to 48%. But the Ipsos Mori survey for the London Evening Standard is unlikely to be enough to allay prime ministerial nerves, with other polls in the last 24 hours showing a lead for Leave.
With no exit polls being conducted by broadcasters, it is likely to be well into the early hours of Friday before a reliable picture emerges of how Britain has voted in what Mr Cameron has described as the most important vote in a lifetime. And the final result is not expected to be confirmed at the Electoral Commission's main counting centre in Manchester until breakfast time.
A record number of voters are eligible to take part in the referendum, with the Electoral Commission putting the number at 46,499,537.
Anecdotal evidence from around the country suggests that turn-out has been high in many areas - something widely expected to favour the Remain camp. Fine weather in Scotland heartened Remain campaigners, who are relying on a high turnout north of the border, where voters are thought to be strongly pro-EU.
But torrential rain and flooding in the South East caused transport disruption which may have prevented some voters from reaching the ballot box in time. Some polling stations were forced to close, and two in Kingston-upon-Thames had to be relocated after becoming inundated.
Leave campaign standard-bearer Boris Johnson had a last-minute dash to vote in north London, due to a delay to his flight from Scotland after attending his daughter's university graduation ceremony - finally reaching the polling station with less than 25 minutes to spare.
Waiting for his plane in Edinburgh, he told reporters that polls had been "very close" but turnout was "good in areas where we need it to be".
Prime Minister David Cameron ignored questions as he arrived with wife Samantha to cast their votes at Methodist Hall in Westminster, saying only "Good morning" to the gathered media from across the world.
Boris Johnson only just made it in time to cast his vote - arriving just 25 minutes before the polls closed.
He had been attending his daughter's graduation in Edinburgh earlier in the day, but ran into difficulty when his flight back to London was delayed.
The former mayor of London arrived at his polling station in Islingston, north London, with his wife Marina Wheeler, to some heckles from the crowd.
But Mr Johnson did not appear stressed, telling the Press Association he was "not at all" worried about not making it in time.
Asked how he felt about the vote he said: "Let's see, let's see. It's in the hands of the people now."
He added that the graduation had been "wonderful", before making the short journey home on foot, accompanied by police officer
Labour leader Jeremy Corbyn - who has been accused of campaigning half-heartedly in the Remain cause - said he was "extremely" confident of the outcome, pointing to bookmakers' odds which have consistently favoured continued EU membership despite fluctuations in the polls.
Pro-Leave Justice Secretary Michael Gove said he was feeling "quite excited" as he accompanied his wife, Sarah Vine, to a polling station in west London.
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 former London mayor Mr Johnson, have urged voters to "take back control" of the country.
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.
