Putin was the president of Russia from New Year's Eve of 1999 until spring 2008. He then took a break and served four years as prime minister, but returned for a third term as president last year.
Putin, former French Prime Minister Francois Fillon and other officials were gathered in Valdai, Russia, for the 10th annual meeting of the Valdai Club, a four-day discussion among Russian and foreign officials, RIA Novosti reported.
Following Putin's speech on his country's national identity, during a public question and answer period, the Russian president asked Fillon if he planned to run for the French presidency. Fillon said he would answer only if Putin responded to the same question, Novosti said.
"I don't exclude [the possibility]," Putin said, prompting Fillon to quip, "And I, too, don't exclude [the possibility]."
Putin has presided over deteriorating relations with the United States over the Syrian civil war and over Russia's granting of asylum to NSA leaker Edward Snowden.