Valls, who quit President Francois Hollande's cabinet to run, was the long-time favourite to win the nomination before campaigning began.
But Valls' bid has been viewed as lacklustre and some pollsters believe it will be two candidates from the party's left flank — protectionist maverick Arnaud Montebourg and Benoit Hamon — who emerge Sunday to contest a
second-round run-off a week later.
The odds will be stacked against the victor, with many polls showing the Socialist candidate will be eliminated in the first round of the presidential election on April 23.
Current projections suggest the election is 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 has also served in the governing Socialist government.
Macron, a relative newcomer to politics who resigned from the government in August to found his own centrist movement, has stolen a march on his Socialist rivals over the past two weeks with speeches packed to capacity.
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 him over their party's nominee if he looks to have a better chance against Le Pen. Asked during a televised debate Thursday whether they would contemplate stepping back and supporting Macron, Valls, Montebourg and Hamon all demurred.
“He's a Tony Blair of 20 years ago,” Hamon said dismissively.
Macron himself has ruled out a pact with the Socialists, announcing Thursday he would field his own candidates in parliamentary elections set for June.
Communist-backed firebrand Jean-Luc Melenchon, who like Macron is polling in double digits, also risks splitting the leftwing vote.
A Fillon-Le Pen presidential runoff is currently seen as the most likely scenario in May, as voters across Europe tilt towards anti-immigration parties like Le Pen's National Front.
Le Pen told a meeting of right-wing populist parties in the German city of Koblenz on Saturday that Europe was about to “wake up”, following the victory of Donald Trump in the US election and Britain's vote to leave the European Union.
“The stakes for the PS are historic,” said former Socialist party strategist Gerard Le Gall, because the election will either reaffirm or undermine the party's supremacy on the left, a position it wrested from the Communists in 1978.
The signs are not good for the Socialists, whose primary has failed to energise voters hungry for change after four years of stagnation under the deeply unpopular Hollande, who decided in December not to seek re-election.
The Socialist party is expecting around 1.5 million voters to take part on Sunday, far fewer than the 4.0 million who turned out for the centre-right Republicans primary.
In the final debate between the seven candidates Thursday, Hamon was attacked over his proposal to pay the poor and 18-25-year-olds a universal income of 600 euros ($641) a month.
Valls, who has portrayed himself as a steady pair of hands, hit out at “leftists who make unlimited promises”: Montebourg noted that Hamon's proposal would take up most of France's budget.
Valls, who was slapped this week by a protester, set out to modernise the Socialist 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.