The poll by Opinium for the 'Observer' showed that the Brexiters now stand at 43 per cent, while 40 per cent say they support the campaign to keep the UK in the 28-member economic bloc in the June 23 referendum.
Brexit, an abbreviation of "British exit", refers to the possibility that the UK will withdraw from the European Union.
The latest poll suggests the remain camp has lost four percentage points in the last two weeks, during which former London mayor Boris Johnson and Conservative MP Michael Gove have relentlessly campaigned on the theme of immigration.
Just over a third (35 per cent) cite Britain's ability to make its own laws without EU interference and 29 per cent cite the impact of leaving on the UK economy.
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In the last Opinium survey two weeks ago, in terms of trends - 55 per cent were leaning towards remain and 32 per cent leaning towards leave.
In the latest survey, the gap has narrowed dramatically, with 36 per cent leaning towards remain and 33 per cent towards leave.
Half of the 2,007 people surveyed said they believed that immigration would be under better control if the UK did leave the EU.
The leave campaign also believes that a relatively poor turnout among Labour voters who support remain could deliver it victory.
Meanwhile, former British Prime Minister John Major launched a withering attack on the Vote Leave campaigners for running a campaign of "deceit".
"They are misleading people to an extraordinary extent. They are feeding out to the British people a whole galaxy of inaccurate and frankly untrue information. And what they have not done is tell us what would be the position if we were to vote to leave,"he told BBC today.
"If we quit the EU the country would be poorer, there would be volatility in the financial markets and that would push up mortgage costs irrespective of what the Bank of England might do with official interest rates," UK Chancellor George Osborne told 'The Sunday Times'.