Al Jazeera, citing Iranian news outlet Khabar Fouri, reported that the tanker was struck by four US missiles and that an evacuation operation was underway.
Kharg Island is strategically important to Iran's oil industry and handles a major share of the country's crude oil exports.
There has been no official confirmation of the reported missile strike so far. No casualties have been reported.
Fighting between the United States and Iran has sharply escalated following a month-long lull, with the two sides exchanging strikes amid growing concerns over civilian casualties.
The US military recently struck rocket launchers on an island in the Strait of Hormuz, days after US President Donald Trump said American forces had cleared mines from the crucial maritime trade route.
The US said the strikes were intended to prevent Iran from laying new mines in the waterway.
Meanwhile, Iranian authorities said US strikes hit a wedding party in Kuhestak on Wednesday, killing at least four people and injuring 68 others.
The US military said it was investigating the incident and reiterated that it does not target civilians.
The latest exchange represents a major escalation in the conflict, which began in February after the United States and Israel launched military operations against Iran.