Thirty-eight per cent of those polled by YouGov for 'The Times' backed "Brexit" - or an exit from the 28-nation economic bloc - and 37 per cent wished to remain, with 25 per cent undecided or not planning to vote.
If the undecided voters are excluded, it means 51 per cent are in favour of Brexit and 49 per cent want to stay.
It indicates that British Prime Minister David Cameron's deal struck in Brussels last week may have swayed a considerable number, with those in favour of leaving the EU dropping considerably from previous months.
When the question was last asked, 17 per cent thought they would be better off and 16 per cent worse off.
The poll comes as the UK's most senior former military commanders signed a letter to 'The Telegraph' warning that Britain must stay in the EU so it can protect itself from "grave security threats" caused by Islamic State (ISIS).
The letter reads: "We have served around the world and in almost every conflict in which Britain has been engaged since the Second World War.
"Britain will have to confront these challenges whether it is inside or outside the EU. But within the EU, we are stronger. Inside it, we can continue to collaborate closely with our European allies, just as we did when we helped to force the Iranians to the negotiating table through EU-wide sanctions, or made sure that Putin would pay a price for his aggression in Ukraine."
The campaign in the lead up to the referendum is gathering momentum with Cameron launching a national tour yesterday to make the case for staying in the EU.
Nearly 95 per cent of bosses polled by the Confederation of British Industry and UK's real estate firm CBRE said that London's access to European markets was its biggest strength.
On the other side on the argument, the "Vote Leave" campaign has claimed that the country will be better off outside the union.
"Voting to leave the European Union would be a massive boost to UK-India relations. New opportunities for the UK and India to cooperate more closely and develop stronger trading links would emerge as the UK re-aligns its foreign policy and trade priorities," Priti Patel, Cameron's Indian Diaspora Champion, said in a statement yesterday.
