One militant group had issued explicit direct threats to disrupt the Olympics, a State Department official said.
Two bombings in the Russian city of Volgograd over the last two days, one at the city's central railway station and another on a bus, killed dozens of people and raised anxieties about the safety of the Olympics.
U.S. officials said U.S. and Russian authorities have engaged in extensive contacts regarding security preparations for the Games. The United States is expected to share with Russia information it might collect about possible threats to the Games.
"We're taking lots of security precautions" related to the Winter Games, a U.S. State Department official said on Monday.
U.S. security officials said the government was not surprised by the Volgograd bombings and had anticipated that such attacks might well occur in the run-up to the Games.
Caitlin Hayden, a spokeswoman for the National Security Council, condemned the attacks in Volgograd blamed on suicide bombers.
She added that the U.S. government had "offered our full support to the Russian government in security preparations for the Sochi Olympic Games, and we would welcome the opportunity for closer cooperation for the safety of the athletes, spectators, and other participants."
The State Department is expected to caution U.S. travellers on Monday about possible bombings and hostage takings in Russia, particularly in the North Caucasus region which is less than 100 miles (160 km) from Sochi.
The U.S. offer for closer cooperation with Russia follows two issues earlier this year that raised tensions between U.S. and Russian security agencies: the involvement of two Chechen brothers in the Boston Marathon bombing and Russia's granting temporary asylum to former U.S. National Security Agency contractor Edward Snowden.
Despite these points of contention, a U.S. official said security cooperation between the U.S. and Russia regarding the games had improved in the run up to the Olympics.
In November, Matthew Olsen, director of the government's U.S. National Counter-terrorism Centre, said his agency was "coordinating and integrating the intelligence community's support...to the Winter Olympics in Sochi."
Olsen told a Senate committee he had visited Sochi a few days earlier and met Russian intelligence and security officials to discuss the games.
(Editing By Alistair Bell and Andrew Hay)
