Vladimir Putin tried to calm one leopard cub, which had become agitated and attacked two journalists, Russian media said.
The visit to Sochi National Park comes amid efforts to show the games are positive for the environment.
Vladimir Putin is well known for his encounters with wild animals which have helped burnish his outdoors image.
In the past, the president has been shown on state TV tagging whales, saving a TV crew from a tiger and taking to the skies to fly with Siberian cranes.
Vladimir Putin drove Jean Claude Killy, former French Olympian and International Olympic Committee (IOC) official, and IOC Executive Director Gilbert Felli to the nature reserve near the Black Sea resort of Sochi.
But a six-month-old Persian leopard cub, Grom (Thunder), attacked two journalists inside a cage, scratching one on the hand and biting the other on the knee, RIA Novosti news agency said.
Footage broadcast on state TV showed the Russian president cuddling the leopard seemingly at ease, commenting: “I like animals, it seems I have a feeling for them. We liked each other.”
Russia is working to re-introduce the Persian leopard to southern Russia where they became extinct in 1970.
“We’ve decided to restore the population of the Persian leopard because of the Olympic Games,” Vladimir Putin said.
During the visit Vladimir Putin insisted that the enormous construction effort to prepare Sochi for the Games had greatly improved the environment.
Environmentalists have accused the Russian authorities of damaging the environment beside the sea and in the mountains of the Sochi area during construction for the Games.
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