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A YouGov 'on-the-day poll' suggests 52% for Remain and 48% for Leave.
ITV News results analysis suggests there is a 75% probability that Leave will win.
"Leave are doing better than we expected them to do than two or percentage points," our elections analyst Colin Rallings said.
Professor Jane Green said Remain need to do far better in upcoming results to stand a chance of winning.
He said smaller margins of victory and weaker turnout than expected in Scotland was having an impact.
ITV News predicts Leave will win the EU referendum.
"An extraordinary moment in British history," News At Ten host Tom Bradby said.
The Chancellor George Osborne has said the UK economy is "about as strong as it could be", as he looked to calm market fears following Brexit.
Speaking as financial trading started on Monday, Mr Osborne said the Leave vote in last week's EU referendum was "not the outcome I wanted" but said he "will do everything" he can to "make it work for Britain".
He added that "there will be an adjustment in our economy because of the decision that the British people have taken" but made no indication that there would be an emergency budget.
"I respect that decision and we're going to get on and deliver on that decision. But the impact on the economy will have an article on our public finances," he said.
His comments came after the pound suffered fresh losses during early trading on Asian markets on Monday morning.
Addressing the question of the free movement of labour following withdrawal from the EU, the chancellor said Britain is an "open and tolerant country" and he "will fight to keep it so".
Matt Rooney from Nottingham is on holiday in Greece. When he arrived at reception this morning wanting to exchange British currency for Euros he was greeted with a sign saying they weren't able to because they don't have an "official exchange rate from the central bank".
He tweeted this picture - his tweet has been retweeted 9,400 times.
The problem, which happened after the EU Referendum result was announced, has now been resolved.
Hear more from him on our programme at 6pm.
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