Russia's deputy military chief Anatoly Nogovitsyn said the NATO warships' entrance to the Black Sea is a "serious threat to our security," Hurriyet daily reported on Thursday.
He said under the Montreux Convention, signed in 1936 on the status of the Turkish Straits, the warships can only stay in the Black Sea for 21 days.
"If the NATO ships continue to stay in the Black Sea after the expiration of 21 day-period, then I would like to remind you that Turkey would be responsible," he added.
The U.S. warships are spearheading a humanitarian aid mission to Georgia, a U.S. ally that wants to join the North Atlantic Treaty Organization.
Clashes erupted in the Caucasus after Russia responded to Georgia's military operation to regain the control in the breakaway region South Osstia.
The U.S. ships are carrying nuclear missiles that can hit Russian targets as far away as St. Petersburg, Nogovitsyn said, according to Hurriyet. Russia has dispatched its own ships to track the U.S. vessels, the newspaper said.