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 runoff.
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.
"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.
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