The so-called Brexit side edged ahead in a key survey just over a week before Britain votes on its future within the economic bloc next Thursday.
A YouGov survey for The Times puts "Leave" on 46 per cent, up three points since the end of last week, and "Remain" is on 39 per cent, down three - giving Brexit its largest lead since the start of the campaign. 11 per cent of people do not know how they will vote and 4 per cent plan to abstain.
An ICM poll for TheGuardian yesterday gave the Brexit campaign a six-point lead, with Leave on 53 per cent and Remain on 47 per cent, while an ORB poll for The Daily Telegraph has put "Leave" on 49 per cent among those certain to vote on June 23, one point ahead of Remain at 48 per cent.
John Mills, chair of the Labour Leave campaign, said "The polls are swinging our way, which is very good news but we must not be complacent".
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"The Leavers' false claim is that, if we leave, there can be higher non-EU immigration while bringing down overall numbers. Let's not be seduced by this lie from people who are anti-immigration and who have spent their lives campaigning against the interests of working people," they wrote in the Guardian today.
In a direct attack on Conservative party's most prominent Indian-origin voice in favour of Brexit, Priti Patel, they add "Where there are shortages in certain sectors, like our curry houses, it's government rules that are the root of the problem".
The British Election Survey (BES) released last month had found that 51.7 per cent of Indian-originvotersare against Brexit, compared to 27.74 per cent in favour of leaving the economic bloc.
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Opinion polls are not being seen as a reliable reflection of the mood of the voters by either side ever since they had decisively predicted a hung Parliament in the May 2015 general election which threw up one of the largest Conservative party majorities in history.
The Rupert Murdoch-owned tabloid is famous for mostly backing the winning horse in elections in the UK.
In a front page editorial under the headline "BeLeave in Britain", the paper warned that staying in the EU would be worse for immigration, worse for jobs, worse for wages and "worse for our way of life".
"This is our last chance to remove ourselves from the undemocratic Brussels machine and it's time to take it," it said.
A legal action brought by the European Commission against the British government over planned tests for foreigners to claim benefits was thrown out by Europe's highest court.
Britain had a right to impose a more stringent test for child benefit and child tax credit - known as a "right to reside" - for EU migrants in order to protect its finances,the court said.
Aspokesperson for HM Revenue and Customs said, "The UK welcomes the Court of Justice of the European Union's judgement, which supports our view that we are entitled to ensure only EU migrants who have a right to be in the UK can claim our benefits".