The Thai navy said the vessel is owned by a Thai transport company, Precious Shipping.
The company said it is working with the relevant authorities to rescue the three crew members, while the 20 rescued sailors have been safely evacuated and are now ashore in Oman.
It added that the vessel was struck by two projectiles.
Iran has launched strikes against its oil-exporting neighbours, threatening shipping in the Strait of Hormuz and plunging the global energy economy into crisis.
It was not immediately clear whether the incident was one of three commercial ships that the United Kingdom Maritime Trade Operations centre earlier said had been hit in the Gulf on Wednesday.
Ship-tracking websites showed the Mayuree Naree just off the Omani coast in the Strait of Hormuz, moving slowly at little more than one knot.
The carrier is 178m long and displaces 30,000 tonnes, they said, adding it was on its way to Kandla in India.
AFP sought comment from the firm in Bangkok but did not receive an immediate response.