Dozens of people were queuing in Caracas neighborhoods including Chacaito and Los Palos Grandes before polling stations opened at 7:00 am (1100 GMT), according to the Democratic Unity Roundtable (MUD) opposition coalition.
They are due to close at 4:00 pm (2000 GMT), though they will remain open as long as people are in line.
Delegates and volunteers, many dressed in white, manned tents and tables at some 14,300 polling stations nationwide.
Opposition leaders expect as many as 11 million of people to cast ballots anyway, voting to reject the president's controversial plan for a separate referendum July 30 to elect a constituent assembly to rewrite the constitution.
The opposition is boycotting the Maduro-backed vote.
They hope a big turnout Sunday will increase pressure for Maduro's removal from power, clearing the way for new presidential elections before his term ends in January 2019.
The rival elections have given rise to international worries -- voiced by the Catholic Church and UN Secretary General Antonio Guterres -- that the chances of bringing both sides together for dialogue have become more remote.
Maduro portrayed Sunday's vote as merely an "internal consultation by the opposition parties" with no electoral legitimacy.
But he also urged Venezuelans to "participate peacefully."
While Maduro is deeply unpopular -- with 80 percent of Venezuelans criticizing his rule, according to the Datanalisis survey firm -- he enjoys backing from some, mostly poor, parts of the population and, most importantly, from the military.
Many Venezuelans, though, are less focused on the political power play than they are on just getting by day to day under a crushing economic crisis that has led to shortages of food and medicine.
Opposition figure Maria Corina Machado predicted the vote would not only reject the Constituent Assembly but also "give a mandate for a change of the regime.
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