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The two leading candidates, François Fillon and Marine Le Pen, are both avowed opponents of sanctions meant to punish Moscow for its annexation of part of Ukraine and its support for rebels in the country's east.
Russia's bombing of Syria's onetime commercial capital of Aleppo—called a war crime by France's current leaders—hasn't deterred either politician from urging closer ties between Paris and Moscow.
The victory of either candidate in the May election threatens to blow a hole in Europe's sanctions against Moscow, which are a centerpiece of the Continent's strategy for containing Russia's military assertiveness.
With U.S. President-elect Donald Trump also promising friendlier relations with Moscow, Western agreement on sanctions against Russia could crumble.
The rise of pro-Russia leaders in France could also soften European opposition to Syrian President Bashar al-Assad.
Under President François Hollande, France has led calls for the Syrian leader to step aside. That has escalated tensions with Mr. Putin, who has deployed forces to keep Mr. Assad in power.
Shortly before Mr. Fillon swept to victory in France's center-right presidential primary last month, the 62-year-old former prime minister delivered a stinging critique of the sanctions.
"Is it intelligent, with a sanctions policy that has done absolutely nothing but ruin French farmers, to push Russia both into diplomatic mistakes and at the same time toward Asia?" Mr. Fillon said in a debate, referring to retaliatory import bans Moscow has imposed on European farm exports. "I think this policy has failed."
Since Russia annexed Crimea in March 2014, German Chancellor Angela Merkel and Mr. Hollande have led the push to impose several rounds of European Union sanctions on Moscow.
Needing unanimous support from the other 26 EU nations, the pair was able to rally national capitals to back the measures, overcoming pockets of skepticism across Europe.
With a sanctions opponent in Paris, the measures would be at risk of expiring in July, when EU nations must vote to renew them.
Europe may have an additional incentive to drop the measures if Mr. Trump decides to scrap U.S. sanctions, which have largely mirrored those adopted by the EU.
Of the two candidates, Ms. Le Pen is the more strident supporter of Mr. Putin.
Mr. Fillon, however, is more likely to become president. Polls show him winning easily in a runoff against Ms. Le Pen. Diplomats across Europe are now trying to figure out what Mr. Fillon would actually do once in office.
"There are different shades of 'Russia friendliness' in France," said one EU diplomat. "Compared to Marine Le Pen, Fillon isn't that bad."
Aides to Ms. Le Pen didn't respond to a request to comment.
Caroline Morard, a spokeswoman for Mr. Fillon, said the candidate maintains the economic sanctions should be lifted "as soon as possible."
France's relationship with Germany would loom large in Mr. Fillon's calculations, analysts say. Mr. Fillon has said he would make a priority of strengthening relations with Berlin, where Ms. Merkel has labored to maintain European unity behind the sanctions.
"It's difficult to see a Fillon administration coming into office and just leaving Germany out in the cold," said David Cadier, a researcher at the Johns Hopkins School of Advanced International Studies.
And, unlike Mr. Trump or Ms. Le Pen, Mr. Fillon hasn't gone as far as questioning the North Atlantic Treaty Organization, Europe's main defensive bulwark.
Mr. Fillon has spent years quietly cultivating a relationship with Mr. Putin. The two met frequently when Mr. Fillon served as prime minister under then-French President Nicolas Sarkozy, starting in 2007. In 2013, Mr. Fillon was guest of honor at a policy gathering hosted by Mr. Putin in Russia.
"It is through our dialogue, dear Vladimir," Mr. Fillon said at the gathering, in a speech discussing the conflict in Syria, "that we will find peace."
Relations with Russia have become a heated campaign issue in France.
Alain Juppé, Mr. Fillon's main opponent in the center-right primary, warned this fall about Mr. Fillon's ties to Mr. Putin, saying: "Be careful of too much vodka."
France's center-left contenders for the presidency have also chided Mr. Fillon for his stance on Russia, though they are running far behind in the polls.
Mr. Fillon has recently sought to minimize his personal relationship with Mr. Putin, insisting that the two aren't friends. He has cast his position on Russia as one of pragmatic necessity, arguing in effect that the country is too large and powerful to be antagonized.
"It's a nation full of nuclear weapons," he said during last month's debate. "It's a dangerous country if we treat it the way we have the last five years."
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