Iran Press/ Europe: First projections of results are expected at 8 p.m. (1800 GMT/UTC), which is when the final polling stations are scheduled to close.
Preliminary official results are then expected later on Sunday evening.
Voters in France's overseas territories cast ballots earlier in the weekend.
Some 49 million people are eligible to vote.
The election was called by President Emmanuel Macron after his party suffered a resounding defeat at the hands of the far-right National Rally (RN) in European Parliament elections early in June.
Polls suggest that the National Rally has a chance at winning a parliamentary majority, which would put France under the control of far-right forces for the first time since the Nazi era.
The election is also being contested by a new leftist alliance, the New Popular Front (NPF), recently established tocontest the elections.
The Ensemble (Together) grouping led by Macron's Renaissance party was trailing in third place in preelection surveys, ahead of the conservative Republicans (LR).
Macron has a presidential mandate until 2027 and has said he won't step down before the end of his term, even if he has to contend with a prime minister who holds radically different ideas from his own.
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