The tanker was one of two vessels that were rocked by explosions as they passed through Gulf of Oman waters on Thursday.
The Front Altair is "heading toward the Fujairah-Khor Fakkan area in the United Arab Emirates", head of ports of Iran's southern province of Hormozgan told the semi-official news agency ISNA.
The tanker "has left Iran's territorial waters," he said, adding that it was being towed and sprayed with water to cool the hull. The Norwegian company which owns the ship confirmed it had left Iranian waters and said specialists would board the vessel on Saturday to assess the damage.
"The 23 crew members of the 'Front Altair' remain safe and well, with plans being made for their repatriation soonest," Frontline Management said. T
he crew, who were rescued by Iran after the vessel caught fire and transferred to Bandar Abbas, were to fly to Dubai on Saturday night, ISNA said.