The result will be decided based on which campaign gets most supporters and how floating voters divide, according to polling experts.
Around one in 10 voters remains undecided despite months of campaigning and could only decide when he enters the polling booth.
Older voters are more likely to back Brexit, with almost twice as many people aged over 65 set to vote for leaving the EU than those under 35.
There were also signs that those backing Brexit (the common term used for Britain's exit) are more energised to vote, with the 'Out' campaign given a seven-point lead by one poll.
Britain's most trusted polling expert said there was little sign of the swing back towards staying in the EU that had been expected in the last week.
Professor John Curtice, President of the British Polling Council whose exit poll at the 2015 general election revealed the Conservatives would win a majority, said there was little gap between the sides.
"(Remain) has grabbed some of the ground that it seemed to have lost in last week's polls -- but it's certainly not grabbed back all of the ground," he told BBC Two's Daily Politics programme.
He added that "so far at least there isn't any clear evidence" of the swing back to the 'Remain' vote.
The new polls suggest that trends in recent months remain unchanged as Britain votes in what has been dubbed the biggest decision the country will take for a generation.
Opinium Research's final poll before the referendum, which asked around 3,000 people for their views, put 'Leave' on 45% and 'Remain' on 44%.
Adam Drummond, a polling analyst for the company, said: "This really is 'too close to call' territory with undecided voters holding the balance of the vote in their hands."
However, two polls released on Wednesday evening put the 'In' campaign ahead. YouGov put 'Remain' on 51% and 'Leave' on 49%, while ComRes had the numbers at 54 and 46 - an eight-point in lead.
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