The online poll of 1,198 people, carried out on April 12-14, showed support for the "In" campaign led by Prime Minister David Cameron had risen to 38 percent from 35 percent in a similar poll carried out on April 7-11.
Support for the "Out" campaign fell to 34 percent from 35 percent over the same period while the proportion of undecided voters fell to 28 percent from 30 percent over the period, according to TNS.
"After several weeks of no movement in the voting intention, our latest poll shows a four-point lead for the Remain camp," said Luke Taylor, TNS's Head of Social and Political Attitudes.
"It will be interesting to see what effect, if any the now officially appointed campaigns have in the coming weeks and whether either side begins to capture the still very significant chunk of undecided voters," Taylor said.
When respondents were asked how they thought Britain would vote, 40 percent said they thought Britain would opt to stay, 26 percent said voters would seek an exit and 34 percent said they did not know.
A British exit from the EU would rock the Union - already shaken by differences over migration and the future of the euro zone - by ripping away its second-largest economy, one of its top two military powers and by far its richest financial centre.
Pro-Europeans, including former prime ministers Tony Blair and John Major, have warned that an exit could also trigger the break-up of the United Kingdom by prompting another Scottish independence vote if England pulled Scotland out of the EU.
Members of Britain's 'Out' campaign say such warnings are overblown and that Britain would prosper if it broke free from what they say is a doomed German-dominated bloc that punches way below its weight beside rivals such as Russian President Vladimir Putin.
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