The findings surprised many commentators ahead of the June 23 referendum because most recent polls have given the "Remain" camp led by Prime Minister David Cameron a narrow lead.
One of the ICM polls for the Guardian newspaper was conducted online -- where pollsters typically believe more voters say they want to leave the EU -- and one was conducted by telephone.
The online vote showed 47 per cent would like to leave while 44 per cent backed Prime Minister David Cameron's "Remain" campaign. A further nine percent were undecided ahead of the June 23 vote.
It was only the third phone poll in the campaign to put the "Leave" camp ahead.
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"Our poll rather unhinges a few accepted orthodoxies," ICM's director, Martin Boon, told the Guardian.
"It is only one poll, but in a rather unexpected reverse of polling assumptions so far, both our phone poll and our online poll are consistent".
Once undecideds were excluded, both new polls gave 52 per cent support for Brexit compared to 48 per cent for "Remain".
"Leave" campaigners including former London mayor Boris Johnson last week seized on figures which put net migration into Britain at 333,000 last year to argue that it should pull out of the EU.
A poll of polls of the last six surveys maintained by the What UK Thinks academic project currently puts "Remain" on 51 per cent and "Leave" on 49 per cent.