The US Navy denied that its operations in the region were affected.
The Iranian navy chief, Adm Habibollah Sayyari, was quoted as saying that the US warship received two warnings before leaving the region.
"They were warned once by maritime patrol aircraft and once again by the Alborz destroyer," he said. "The US warship left the region immediately after receiving the warning," he added, saying he believed that the Americans had approached the area to "learn our tactics."
It is the first Iranian naval exercise since 10 US sailors were briefly captured by Iran earlier this month after drifting into Iranian territorial waters.
Cmdr Kevin Stephens, a spokesman for the US Navy's 5th Fleet, offered a different version of events.
Stephens said the US is aware of the Iranian exercise and that it has not altered US naval operations in any way. Iran had announced plans to close off certain areas for the drill, but he said this is "common practice for any navy conducting such training at sea," he said.
Iran conducts similar exercises annually and the US does not see this year's as a change in Iranian behavior, Stephens added.
He said the exercise may increase the chances of US forces interacting with Iranian warships, but added that US Navy vessels are routinely approached by Iranian warships as they operate in the region.
Most of those encounters are conducted "in a safe and professional manner, recent events notwithstanding," he said.
The 5th Fleet is the US Navy's regional counterweight to Iran and is based across the Gulf from Iran in the island kingdom of Bahrain.
Iran occasionally holds military maneuvers to ensure the preparedness of its forces and to demonstrate its military capabilities in the region. The county has more than 2,000 kilometers of shoreline facing the Persian Gulf and the Sea of Oman.
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