Sochi is hosting the Winter Games in February, a pet project for President Vladimir Putin, who is determined for them to be a success. The overall bill for the games stands at $51 billion, making them by far the most expensive Olympics in history.
Doku Umarov, a widely known Chechen rebel leader, urged his fighters to "do their utmost to derail" the games, which he described as "satanic dances on the bones of our ancestors."
"We have the obligation to use all means to prevent this," he said in a video posted on a rebel website on Wednesday.
Umarov last year urged his fighters to avoid hitting civilian targets because Russians in Moscow were taking to the streets en masse to protest against Putin.
Security experts have said the Islamic insurgency raging across the North Caucasus mountains that tower over Sochi is a daunting threat to the games—although rebels have not attacked Sochi so far.
Dagestan, which lies about 500 kilometers (300 miles) east of Sochi, has become the center of the insurgency that spread across the North Caucasus region after two separatist wars in the 1990s in neighboring Chechnya. Rebels seeking to carve out a caliphate, or Islamic state, have targeted police and other officials in near-daily shootings and bombings. Umarov is believed to be their most influential leader.
Tamerlan Tsarnaev, the elder of the two ethnic Chechen brothers who are accused of staging the Boston Marathon bombings, spent six months last year in Dagestan.
Jean-Claude Killy, head of the IOC coordination commission for Sochi, said such threats are to be expected.
"We get threats before every Olympics," he told The Associated Press. "This cannot be taken lightly. I think the Russians are well equipped to face the challenge."
The Sochi 2014 Organizing Committee said in an emailed comment that ensuring security at the games is "the responsibility of the state" and will be its priority.
"We are confident that the games will be safe and comfortable for all as guaranteed by the Russian state," the committee said.
Russia's National Anti-Terrorism Committee said in a statement on Wednesday that it is working to eliminate threats at all international sporting events in the country including the Olympics. Officials said that they aim "to identify and avert various threats, including those of terrorism."
Ramzan Kadyrov, a strongman who rules Chechnya, played down the influence of Umarov and promised that his security forces would track him down. "Before the Olympics, I think, I'm sure, that we will destroy him," Kadyrov told the Interfax news agency. "We search for him every day, but he is nowhere to be found."
Press officers for the Interior Ministry in Russia's southern district, which includes the North Caucasus, declined to comment.
An officer of the Interior Ministry's special task force in the North Caucasus, who asked not to be identified because he was not allowed to comment publicly, told The Associated Press that Umarov's statement could prompt Russians to step up their efforts to comb the mountainous areas where Umarov could be hiding.
The United States declared Umarov's group a terrorist organization in 2011 and offered up to $5 million for information leading to his capture.
Umarov's group is blamed for bombing a Moscow airport in January 2011, two subway stations in 2010 and a Russian train in 2009.
AP writer Stephen Wilson in Lausanne, Switzerland, and Sergei Venyavsky in Rostov-on-Don, Russia, contributed to this report.