Marine Le Pen's anti-immigration party National Rally came out on top of first-round voting last Sunday, followed by a coalition of centre-left, hard-left and Greens parties – and President Emmanuel Macron's centrist alliance in a distant third.
The second-round voting began Saturday off the Canadian coast in the North Atlantic territory of Saint-Pierre-et-Miquelon, and follows in French territories in the Caribbean, South Pacific and the Indian Ocean, along with French voters living abroad.
The elections wrap up on Sunday in mainland France.
Initial polling projections are expected when the final voting stations close at 8pm Paris time, with early official results expected late on Sunday and early Monday.
Macron called the snap legislative vote after the National Rally won the most votes in France in European Parliament elections last month.
The party, which blames immigration for many of France's problems, has seen its support climb steadily over the past decade and is hoping to obtain an absolute majority in the second round.
That would allow National Rally leader Jordan Bardella to become prime minister and form a government that would be at odds with Macron's policies on Ukraine, police powers and other issues.
Pre-election polls suggest that the party may win the most seats in the National Assembly but fall short of an absolute majority of 289 seats.
That could result in a hung parliament.
Macron has said he won't step down and will stay president until his term ends in 2027, but is expected to be weakened regardless of the result.
