The campaign run by former PM Manuel Valls has been seen as lacklustre. Photo: Philippe Lopez/ AFP
The primary is being viewed as a crucial test of the party's ability to survive and even re-invent itself, with Socialist President Francois Hollande deeply unpopular after five years in office.
After Hollande ruled himself out of the race, Valls quit his cabinet and was the favourite to win the nomination when the seven candidates began campaigning.
But his bid has been viewed by some observers as lacklustre and two contenders from the party's left flank — protectionist maverick Arnaud Montebourg and Benoit Hamon — will push him hard to reach next Sunday's
The odds will be stacked against the victor, with many opinion polls showing the Socialist candidate will be eliminated in the first round of the presidential election on April 23.
The election appears to be shaping up as a three-way battle between conservative ex-premier Francois Fillon, far-right leader Marine Le Pen and Emmanuel Macron, a 39-year-old ex-economy minister who is outpacing his former Socialist government colleagues.
With voters across Europe moving to the right, most polls currently show a Fillon-Le Pen runoff is the most likely scenario in May.
National Front leader Le Pen told a meeting of rightwing populist parties in Germany on Saturday that Europe was about to “wake up” following the victory of Donald Trump in the US election and the British vote to leave the European Union.
One voter in southwest France said he had voted for Hamon, who has proposed to pay the poor and 18 to 25-year-olds a “universal income” of 600 euros ($640) a month.
“I voted for Benoit Hamon because to me he is the one best placed to redress the Socialist party,” said Jean Claude, speaking in the small town of Millau.
Dominique, a voter in his 40s who cast his ballot in eastern Paris, said he had opted for Valls.
“My main concern is that the left reaches the second round (of the presidential election). Valls is the most credible option against Macron,” he said.
Macron, a relative newcomer to politics, resigned from the government in August to set up his own centrist movement and his speeches have been packed to capacity in recent weeks.
A poll published Thursday gave Macron between 17 and 21 percent of the vote in the first round of the election.
“The Macron effect is real,” said political analyst Stephane Rozes.
Some Socialist heavyweights have hinted they could support Macron over their party's nominee if he looks to have a better chance against Le Pen. Asked in a TV debate Thursday whether they would contemplate stepping aside and backing Macron, Valls, Montebourg and Hamon all demurred.
Macron himself has ruled out a pact with the Socialists, announcing Thursday he would field his own candidates in parliamentary elections in June.
The Socialist primary has failed to energise voters who are hungry for change after economic stagnation under Hollande. Turnout at the halfway point of the day was far lower than in the centre-right Republicans primary last year, party officials said.
Just 400,000 people had voted according to figures from 63 percent of the polling stations. At the same point in the Republicans' primary, the figure was more than twice as high. Voting ends at 1800 GMT.
Communist-backed firebrand Jean-Luc Melenchon, who like Macron is polling in double digits in his campaign as an independent, also risks splitting the leftwing vote.
Valls, who was slapped this week by a protester, set out to modernise the party but has struggled to unite his camp, with his rivals accusing him of betraying leftist ideals by forcing through labour market reforms.
The four other candidates running in the primary are former education minister Vincent Peillon, ecologist Francois de Rugy, ex-MEP Jean-Luc Bennahmias and radical left candidate Sylvia Pinel.