An Ipsos MORI survey shows 52 per cent wanting to stay in the European Union, against 48 per cent for a Brexit.
The final poll conducted before polls opened was by Populus for the Financial Times.
The survey of 4,700 people was conducted right up to midnight on Wednesday night, and put Remain on 55%.
The bookies have consistently given Remain a bigger lead in the betting than they have in the polls, although Leave shortened in the EU referendum betting odds last week before easing again, said the mirror.co.uk.
Polls close at 10 p.m., but the future of Britain's relationship with the EU will come in during a frantic three-hour period on Friday, between 4 a.m. and 7 a.m.
By breakfast time, the result of the EU referendum should be known.
Pollster YouGov will announce the results of its survey once voting has closed on Thursday night - which will be the last indication of which way the vote has swung.
The Electoral Commission has forecast that turnout could be as high as 80 per cent.
A change in the law after chaos at polling stations in 2010 - when some people were denied a vote because of long queues - means that people queuing at 10 p.m. will be allowed to vote, reports the telegraph.co.uk
At the close of the poll, thousands of sealed ballot boxes are collected from schools and church halls which have been doubling up as polling stations and transported to one of 382 counting venues across Britain.
For the count, Britain has been divided into 382 counting areas, which will first verify each ballot, allowing each area to announce the turn out. Then the counting will begin.
The results will be collated and fed by local counting officers to regional counting officers in 12 electoral regions. The results will be announced only when all the areas have concluded their counts.
Unlike at a general election, when MPs only need to win a majority in their constituency to win the seat, in this election every vote counts - which means that the final results for Leave and Remain will creep up slowly as regions make their declarations.
The result will be declared by Jenny Watson, the chairman of the Electoral Commission and the referendum's chief counting officer at Manchester Town Hall.
Among other surveys carried out on poll eve, a survey by ComRes for the Daily Mail and ITV News gave Remain a 6-point lead over Leave - on 48 per cent compared to 42 per cent.
A YouGov survey for The Times, which quizzed 3,766 people online on June 20-22, had each side in a dead heat with 45 per cent.