The campaign to leave the EU took a narrow poll lead just before the referendum, but experts warned that the result remained "too close to call".
Two separate polls showed Leave just ahead despite signs its momentum had stalled.
There were also signs that those backing Brexit are more energised to vote, with the Out campaign given a seven-point lead by one poll when likelihood to turn out was incorporated.
Around one in 10 voters remains undecided despite months of campaigning and might only decide when they enter the polling booth.
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 predicted the Tories 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 the BBC.
The new polls suggest that trends in recent months remain unchanged.
Older voters are more likely to back Brexit, with almost twice as many people aged over 65 set to vote for leaving the EU as those under 35.
More voters think they will be worse off if Britain leaves, while Tory voters are evenly split between voting In and Out.
Opinium Research's final poll, which asked around 3000 people for their views, put Leave on 45 per cent and Remain on 44 per cent.
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."
TNS's final poll before the vote put Leave on 43 per cent and Remain on 41 per cent.
Luke Taylor, head of social and political attitudes at the company, said: "Our latest poll suggests that Leave is in a stronger position than Remain, but it should be noted that in the Scottish Independence Referendum and the 1995 Quebec Independence Referendum there was a late swing to the status quo and it is possible that the same will happen here".
However two polls released later put the Remain campaign ahead. YouGov put Remain on 51 per cent and Leave on 49 per cent, while ComRes had the numbers at 54 and 46 - an eight-point In lead.