The survey of 2,000 people by ORB found that 55 percent believed the United Kingdom should leave the European Union (up four points since our last poll in April), while 45 percent wanted to remain in the European Union (down four points).
Despite these polls, bookmakers are still putting their money on Stay, arguing that the undecided vote will largely vote to remain in the European Union on the 23 of June.
Leave.EU, which has been set up and funded by South Gloucestershire insurance tycoon Arron Banks, is one of the leading groups campaigning for a Brexit vote in the referendum later this month.
Two ICM polls conducted over the weekend - one online and one conducted by telephone - found the Leave vote held 53 per cent support compared with 47 per cent support for Remain. An online poll of 2,001 adults conducted over the same dates put "Leave" at 49 percent and "Remain" at 44 percent.
After two polls early on Monday, putting Leave ahead by between 1% (The Sunday Times) and 10% (The Independent), Betfair's odds on the chances of a vote to remain have now fallen to 68.5%, down from almost 80% last week.
Also throwing support behind the Leave campaign is Britain's biggest-selling newspaper, The Sun, owned by Rupert Murdoch.
"Your constant lying is bad enough but this is despicable, a new low even by your gutter standards", tweeted Peter Coles, an astrophysicist at Sussex University, whose Twitter picture includes the banner "Scientists for EU.U.K".
Prof John Curtice of Strathclyde University pointed out that after the new ICM data, the running average "poll of polls" would stand at 52% for leave and 48% for remain, the first time the Brexit camp has had such a consistent lead. "Polls show it's now too close to call and markets are responding with some very twitchy activity". Both polls showed falls in the share of "don't knows", ICM said.
When turnout is taken into account, a little more than half 56% of those who voted for Labour in last year's general election back remain while 44% support leave.
Those in favor of remaining in the bloc argue that leaving it would risk the UK's prosperity, diminish its influence over world affairs, and result in trade barriers between the United Kingdom and the EU.