The April 16 plebiscite will decide on the biggest change in Turkey's system of governance since the modern republic's foundation almost a century ago, potentially replacing its parliamentary system with an executive presidency.
The survey by pollster Gezici put support for the constitutional change at 51.3 per cent, with "No" votes on 48.7 per cent after the distribution of undecided voters.
The poll was carried out face-to-face with some 1,400 people in 10 provinces on April 8-9. In its previous survey a week earlier it put the "Yes" vote at 53.3 per cent.
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Two other surveys on Wednesday showed the "Yes" vote on 51-52 per cent. The mean average of eight polls collated by Reuters puts the "Yes" vote on 50.8 per cent
The referendum campaign has damaged Turkey's ties with some European allies. Erdogan has described the banning on security grounds of some rallies by Turkish ministers in the Netherlands and Germany as "Nazi-like" tactics.
Voting for Turks living abroad finished on Sunday and Erdogan said this week that those overseas had turned out in greater numbers, a development that pollsters say could benefit him.
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