Updated

EXCLUSIVE: Amid false allegations from Iranian media that a U.S. plane was forced down after accidentally entering Iranian airspace, FOX News learned Tuesday about another tense incident that occurred last month near the Strait of Hormuz.

On Sept. 6, the Iranian Revolutionary Guard threatened to shoot down U.S. helicopters flying cover aboard the USS Peliliu patrolling in the area, according to a classified military transcript of the radio exchange.

According to the transcript, the Peleliu, while en route to the Strait from the Gulf of Oman — was engaged in flight operations using CH-46 helicopters when they encountered an Iranian P3 surveillance plane overhead. The U.S. Navy, which considered the plane as outdated and a non-threatening presence, had a friendly exchange with the Iranians.

The Peleliu established bridge-to-bridge contact with the P3, saying:

USS Peleliu: "Unknown aircraft at 2000ft, this is a coalition warship operation in international waters, we request you remain clear."

Iranian surveillance plane: "Good morning coalition warship, how do you feel?"

But a few hours later, while entering the Strait, the situation gets tense after an Iranian patrol boat demanded the warship's identification number, despite being in international waters.

After the Peleliu identified itself, the Iranian Revolutionary Guard asked if it had any support ships in the area or any helicopters currently flying. The Peleliu responded that it was not authorized to provide that information.

Three hours later, according to transcripts, there was another confrontation.

Iranian Revolutionary Guard: "Your helicopters have breached Iranian air space. You have broken international rules. Your breach has been reported to the Iranian government. You are required to land your helicopters."

USS Peleliu: "No challenges are intended to the territorial integrity and sovereignty of Iran."

Iranian Revolutionary Guard: "Last warning, your helicopters are in jeopardy."

The Peleliu continued without landing the CH-46 helicopters.

Military sources told FOX News that this sort of incident happens nearly daily around the Strait, which is heavily trafficked by oil tankers.

As the only sea passage for the export of oil from the Persian Gulf, the Strait of Hormuz is the world’s most important chokepoint, and any incident there can negatively affect the price of oil.

This incident -- along with January's confrontation in the Strait, where five armed Iranian boats "swarmed" three U.S. Navy warships in international waters are examples of how the U.S. is walking a tightrope in dealing with Iran at a time that there are no relations and no hotline between the two governments that could defuse any sort of crisis. Officials told FOX New that this kind of diplomacy, right now, is being left up to U.S. sailors.

On Tuesday, Iran's semi-official Fars news agency reported that a U.S. plane was forced to land in Iran after it accidentally entered that country's airspace. The Pentagon dened the report and said that all U.S. planes were accounted for.

Soon after, a senior Iranian official denied the Fars report, saying both the aircraft and the people on board were Hungarian.

"The Fars report was not accurate. It was a Hungarian aid plane. No American was on board. The incident happened on Sept. 30," a senior Iranian official told Reuters.

FOXNews' Jennifer Griffin, Justin Fishel contributed to this report.