ORC International surveyed more than 1,000 people Friday and Saturday, finding 57 percent believe a terrorist attack is likely. The poll also found that 54 percent have negative views of President Vladimir Putin and 55 percent of Russia.
The Olympic Torch reached Sochi, a resort on the Black Sea, on Wednesday, two days before the games open.
Matthew Olsen, director of the National Counter-terrorism Center, told the House Intelligence Committee that a terrorist attack is most likely to happen in the Sochi region but not at Olympic venues or the Olympic Village, CNN reported.
"There are a number of specific threats of varying degrees of credibility that we're tracking," he said. "And we're working very closely with the Russians and with other partners to monitor any threats we see and to disrupt those."
He said the most likely organization to strike is Imarat Kavkaz or the Caucasus Emirate, one of a number of Muslim extremist groups in Russia's volatile Caucasus region.
The report did not give a margin of error.
Notable deaths of 2014 [PHOTOS]