"All the ship's crew are safe and the ship is stable too", said NITC, according to AFP's report, which said news of the apparent attack drove prices for Brent crude oil up more than two percent on worldwide markets.
Saudi Arabia had no immediate comment on Friday's reports.
Many observers regard the Yemeni conflict as a proxy war between Sunni Muslim-majority Saudi Arabia and Shi'ite-led Iran. Iranian tankers routinely turn off their trackers as U.S. sanctions target the sale of Iran's crude oil. It turned on its tracking devices late Friday morning in the Red Sea, putting its location some 130 kilometers (80 miles) southwest of Jiddah, according to data from MarineTraffic.com.
The U.S. Navy's Fifth Fleet, which operates in the region, said it was aware of the reports but had no further information.
"Those behind the attack are responsible for the consequences of this unsafe adventure, including the risky environmental pollution caused", Foreign Ministry spokesman Abbas Mousavi told state TV.
Refinitiv ship tracking data gave the Sabiti's last reported position on August 14 as off the southern coast of Iran in the Gulf. Output has since been restored.
Houthi rebels in Yemen said they were behind the attack, but Secretary of State Mike Pompeo said there was no evidence the strikes came from Yemen, but instead they came from Iran, according to the Wall Street Journal. He said the incident was under investigation. It is used by tankers that can not navigate the waterway.