The US military has not confirmed or denied the reports.
IRNA news agency, the country's state-run news arm, identified the drone as an RQ-4 Global Hawk.
RQ-4s typically fly at high altitude to conduct reconnaissance missions.
The drone was flying over one of two crippled tankers in the gulf that the U.S. publicly blamed on Iran.More news: Google Pledges $1 Billion to Tackle Bay Area Housing Crisis
The U.S. military declined to immediately comment if an American drone was shot down but a spokesman said: 'There was no drone over Iranian territory'.
USA concerns about the threat to American forces and interests in the region have increased steadily in recent weeks, particularly after the oil tanker attacks at the entrance to the Gulf on Thursday.
UK Foreign Secretary Jeremy Hunt has backed the United States and laid blame for the attack with Iran. Iran has repeatedly denied the attack.
Hormozgan borders the Strait of Hormuz, where the tanker attacks took place.
Tension between Iran and the United States has risen since previous year, when US President Donald Trump exited a 2015 nuclear deal between Tehran and major powers and reimposed sanctions on the country. The outgoing Secretary of Defence Patrick Shanahan announced the deployment of 1,000 more U.S. troops to the region in response to "hostile behaviour".More news: Shikhar Dhawan ruled out of World Cup, Rishabh Pant confirmed as replacement
An anonymous U.S. official confirmed on Thursday that an American drone was shot down amid heightened tensions between Tehran and Washington over its collapsing nuclear deal.
Earlier, Yemen's Iran-aligned Houthis said they had struck a power station in Al-Shuqaiq city, in the Saudi province of Jizan, with a cruise missile, according to the group's Al Masirah TV.
Furthermore, Washington recently announced the deployment of about 1000 additional troops to the Middle East for "defensive purposes to address air, naval, and ground-based threats" in the region following the attack on the oil tankers. Mysterious attacks also have targeted oil tankers as Iranian-allied Houthi rebels launched bomb-laden drones into Saudi Arabia.
Meanwhile, US Special Representative for Iran Brian Hook heading to the Middle East for meetings in Saudi Arabia, UAE, Oman, Kuwait and Bahrain to discuss "Iran's regional aggression", the State Department said.
"We are closely monitoring the situation and continuing to consult with our partners and allies", Sanders said. White House spokesman Sarah Sanders said President Trump had been "briefed on the reports of a missile strike in the Kingdom of Saudi Arabia". In recent weeks, the Houthis have launched a new campaign sending missiles and bomb-laden drones into Saudi Arabia.More news: Lakers looking to move additional pieces in Davis trade
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