A telephone poll by the Star television channel carried out during voting and the day before gave a 49-54 percent range for 'No' votes against 46-51 percent for 'Yes'.
A similar Mega channel survey, broadcast shortly after polling ended, suggested a 49.5-53.5 percent 'No' vote and 46.5-50.5 percent for 'Yes'.
Defence minister Panos Kammenos said in a tweet after polling closed that the Greeks "proved they don't bow to blackmail, to threats".
"No one can ignore the will of the people to live, to live with determination, to take its destiny into its own hands," Tsipras said as he cast his ballot in his Athens neighbourhood, appearing relaxed and wearing an open-necked white shirt.
But EU leaders have warned a 'No' vote could push Greece on the path to "Grexit" -- crashing out of the eurozone.
Greece's voters -- many of them angry or fearful at capital controls this week that have shuttered banks and rationed ATM withdrawals -- were asked in the referendum whether or not they accepted further grinding austerity in return for European bailout funds.
"We're voting 'No' but we're afraid. But when we vote 'Yes', we're afraid as well. We're afraid on both sides," Nadia, a 63-year-old retired teacher on the island of Poros, near Athens, said after voting, her eyes red from crying.
"When you have to choose between two bad solutions, you choose the least bad, and that's clearly 'Yes'," said Dimitris Kavouklis, 42, as he voted in an upmarket district of the capital.
If a 'Yes' resulted, he said he would resign, and the pressure would be on Tsipras to do the same.
Initial official results from the referendum were expected later Sunday.
Although the referendum offered a long and technical question that many voters grumbled was impossible to understand, minds for the most part were made up, with opinion almost evenly divided.
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