British support for staying in the European Union in a June 23 referendum stood at 38 percent while 34 percent of voters would opt to leave, according to a TNS poll published on Wednesday.
The poll, carried out on April 12-14, showed 28 percent of British voters were undecided. TNS said it polled 1,198 people online.
The survey also said that European citizens were more concerned about the Brexit than the British people themselves.
The survey of five European countries by TNS found 78 percent of Germans, more than two-thirds of Spaniards and almost six out of 10 French wanted Britain to stay in the EU.
The question will be put to a referendum on June 23, when the British public will vote on whether to stay part of the 28 member European Union or strike out alone.
The issue has split the British government, with Prime Minister David Cameron facing viscious opposition from senior figures in his own party such as Justice Secretary Michael Gove and London Mayor Boris Johnson.
Brits currently expect the 'In' camp to win it -- 40 percent expect Britain to stay in the EU, according to Wednesday's survey, compared to 26 percent who expect voters to back leaving.
Their European counterparts were more optimistic, with 47 percent of Germans, 48 percent of Spaniards and 44 percent of Poles predicting Britain will avoid an exit.
Only in France did more people expect Britain to go than not, with 52 percent predicting a Brexit and 48 percent saying the referendum will back staying.
Asked about the consequences of of leaving, only a tenth of Brits said the EU's economy would improve while 38 percent it would deteriorate and 21 percent said nothing would change.
Their continental counterparts were less hopeful, however. Two-thirds of Germans, 43 percent of Spaniards, 39 percent of Poles and 33 percent of French predicted a Brexit would hurt the EU's economy.
Former U.S. Treasury secretaries say Brexit would make Britain poorer and smaller
Eight former U.S. Treasury secretaries said on Wednesday that Britain will be poorer, more inward looking and London's role as a global financial centre could be undermined with the Brexit.
"A vote to leave Europe represents a risky bet on the country's economic future," the former U.S. Treasury secretaries said in an open letter to The Times newspaper.
The letter was signed by Timothy Geithner, Henry Paulson, John Snow, Paul O'Neill, Lawrence Summers, Robert Rubin, Michael Blumenthal and George Shultz. Their warning comes before a visit by U.S. President Barack Obama to visit Britain.
"Brexit could call into question London's role as a global financial centre. For many financial institutions, London has served as the financial springboard into Europe," the former Treasury secretaries said.
"A UK turned inward and less engaged in Europe is less able to lead on the critical challenges Europe faces."
A Brexit would disrupt trade and attempts to renegotiate agreements would be difficult, they said.
"The uncertainty surrounding Brexit already appears to be acting as a drag on investment and market sentiment, and a vote to leave could introduce an extended period of uncertainty that could hinder growth even further," they said.
"The better alternative seems to us to work for progress within the confines of EU membership, without incurring the significant economic risk of Brexit: a smaller, slower-growing British economy for years to come," they said.