Bookmakers are forecasting a 60 per cent victory for those wanting to stay in the EU against 40 per cent for the Leave side.
The figures are based on the most recent betting exchanges between bookmakers.
Their forecast directly contradicts what current opinion polls are saying, with the latest predicting a seven per cent lead for the Leave camp.
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At a news conference in Brussels on Thursday, ESSA, the sports betting trade body, said the betting odds for Brexit should be seen as a more reliable outcome than polls of the 23 June referendum.
However, Eric Konnings, a sports betting integrity officer for the betting company Unibet, conceded that the gap between the Leave and Remain sides had narrowed in recent weeks.
He said, "Six weeks ago, the gap was arguably at its widest with a clear majority for staying but by 14 June the gap had narrowed and is now the closest it has been since bets started on Brexit."
This flies in the face of polls which have seen an increase in support for the Leave.
"However, pollsters are under pressure after getting the outcome of the last UK general election badly wrong."
In that election, pollsters predicted a hung Parliament while the betting markets - correctly - said there would be a Tory majority and victory for UK premier David Cameron.
Konnings said one reason for the relative failings of pollsters in recent years is that the wording of the questions can directly influence the responses they elicit.
"Pollsters can also be biased and it is for these reasons that the betting markets offer an alternative, and perhaps more accurate, way of estimating the outcome of the Brexit vote."
His comments were echoed by political forecaster Williams Leighton Vaughan, of Nottingham Trent University, who said there was "plenty of anecdotal evidence" to point to the accuracy of bookmakers' predictions.
He pointed to a UK parliamentary by election a few years ago where polls gave the Labour candidate a huge 18 per cent lead. The Liberal candidate won, as predicted by bookmakers.
In the 2004 U.S presidential election, bookies correctly called the winner in all 50 states, he said.
More recently, the betting industry also got it right when Scots voted in a referendum on independence in 2015. Pollsters had forecast a tiny win for the "no to separation" while the actual result was 10 per cent in favour of remaining in the UK.
Vaughan said betting odds "always had the Tories winning most seats" in the last UK election, whereas few if any of the pollsters said this.
He told this website, "Betting markets are more accurate than pollsters because they take into account all factors and information, including the polls but also what is happening with the undecided."
"No one can step directly in the future and I am saying is that the opinion polls have got it wrong in the past. Opinion polls provide a valuable source of information but they are only one source and betting and prediction markets provide the most accurate and precise forecasts."
He added, "Voters have no incentive to tell a pollster the truth about their voting preferences or likelihood of voting. But when they are betting with their own money they will tend to think hard about the choices they make using the best information available to them.
"The more money involved, or the greater the incentives, the more efficient and accurate the market will tend to be."
Further comment came from Khalid Ali, ESSA secretary general, who said that, even with the Euro football championships in France at the same time, betting on Brexit was "particularly" healthy.
"A lot of people are placing a bet, mostly but not exclusively, in the UK. In fact, it is for the betting industry the biggest event so far this year."
He added, "All this is significant and matters because of the accuracy of these predictions, as was shown in the last UK general election."