By Jean-Baptiste Vey

PARIS (Reuters) - French Prime Minister Manuel Valls on Sunday raised the prospect he could challenge his boss Francois Hollande as the Socialists' candidate in the 2017 presidential election.

A challenge to President Hollande could divide the Socialists further, just as their center-right rivals get down to serious campaigning against the far right after choosing former prime minister Francois Fillon as their candidate on Sunday.

In past weeks, the message from the Valls camp has been that he would wait for Hollande's decision on whether to seek a second term, and would make a presidential bid of his own only if the president decided to step aside.

However, in an interview published on Sunday in the Journal du Dimanche newspaper, Valls did not rule out running against Hollande in the Socialist party primaries in January.

"I will make my decision in all conscience...Whatever happens, the best interests of the country will influence my decision," he said.

Asked if he was putting pressure on Hollande, Valls replied: "This is a very serious and historic moment for the country. Each of us must be aware of that. I am putting pressure on everyone."

Valls was also openly critical of Hollande's conduct with regard to the publication of a controversial book on his presidency, based on interviews throughout the five-year term conducted by two journalists.

Le Parisien newspaper said Hollande's entourage was "outraged" by Valls's comments and was pressing Hollande to oust him in a cabinet reshuffle.

But people close to Hollande contacted by Reuters denied the president was about to reshuffle his cabinet. "It's completely false," one source close to Hollande said.

Polls suggest no candidate of the left would reach a second round run-off in next May's election, where Fillon would probably face far-right National Front leader Marine Le Pen.

Both pro-business former economy minister Emmanuel Macron and leftist Jean-Luc Melenchon on Sunday continued to rule out taking part in the left's own primaries in January.

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Both have already said they will stand in the presidential election as independents - moves likely to mean that no candidate from the left will reach the all-important run-off stage.

Each would score between 13 percent and 15 percent in the first round of the presidential election, according to a Harris Interactive poll published on Sunday which showed Fillon and Le Pen contesting the second round.

Macron, who himself tested Hollande's authority for months before quitting the government in August, criticized Valls's comments, saying he had gone "too far" and that the competition within the executive was untenable.

Hollande has scored the lowest popularity ratings of modern times in a five-year mandate marked by high unemployment and a series of Islamist militant attacks on French soil.

His authority has also been called into question by the publication in October of the book written by journalists from Le Monde newspaper.

In interviews for the book, Hollande was shown as criticizing many of his own allies, spoke of "problems with Islam" and revealed details of conversations between state leaders.

"We need to state that over the last few weeks, the context has changed. The publication of that book created real disarray amongst the left wing," said Valls, who also warned that the Socialist party risked getting wiped out in next year's vote.

Referring to Valls' interview, Hollande told an audience in Madagascar, where he was attending a gathering for French-speaking countries: "The most important thing is that we all get behind one another."

(Additional reporting by Myriam Rivet; Writing by Sudip Kar-Gupta; Editing by Andrew Callus and Mark Trevelyan)