The USS Jacksonville, a Los Angeles-class sub, "struck a vessel while operating in the Persian Gulf" at about 5:00 am local time (0200 GMT), the statement said.
"The vessel continued on a consistent course and speed offering no indication of distress or acknowledgement of a collision."
Damage to the submarine was limited to one of its two periscopes while the vessel's nuclear reactor "remains in a safe condition" along with the propulsion plant, according to the Fifth Fleet.
A US naval P-3 aircraft searched the area and spotted no debris in the water or vessels in distress, officials said.
Any maritime incident involving US naval ships in the Gulf raises the possibility of a crisis, given tensions between Iran and the United States over Tehran's nuclear program.
Tehran has threatened to block oil tanker traffic through the strategic Strait of Hormuz in retaliation for international sanctions, imposed to push Iran to give up sensitive uranium enrichment work.