Mrs. Clinton laughed off the episode, telling reporters what has become an oft-repeated story about their relationship. “Prime Minister Putin,” she recalled asking him amid his harangue about trade policy, “tell me about what you’re doing to save the tigers in Siberia.”

Eyes brightening, Mr. Putin motioned her to his private office downstairs, where he showed her a map of Russia. Pointing to various regions, he delivered a fervent lecture about endangered tigers and polar bears. He asked her whether Bill Clinton would go on a bear-tagging expedition with him in Siberia. If Bill was not available, he said, maybe Hillary would go?

Diplomats traveling with her were impressed. “It’s like he sizes somebody up and sees them as a worthy adversary or counterpart,” said William J. Burns, a former ambassador to Moscow who served as Mrs. Clinton’s deputy at the State Department. “I’ve seen him with other people who he didn’t see that way, and he’d be much more dismissive and snarky.”

In retrospect, though, the meeting planted the seeds for future tensions. Eight months later, Mrs. Clinton canceled her attendance at a meeting he organized in St. Petersburg to save the tigers. Her excuse was that she had to stay in Washington to lobby the Senate to ratify a new arms reduction treaty.

A year later, Mrs. Clinton was traveling in Lithuania when reports of ballot tampering and other fraud emerged after parliamentary elections in Russia. “The Russian people, like people everywhere, deserve the right to have their voices heard and their votes counted,” she said in a statement, drafted by her spokeswoman, Victoria J. Nuland, a career diplomat and Soviet expert known for her sharp-edged views about Russia.