Minister Christopher Tufton said Bolt was aware of the results and his recent contacts were being traced.
"It is now public knowledge that Mr. Bolt has tested positive. He has been formally notified, I'm told by the authorities, Tufton told reporters Monday evening. "'It triggers an approach to questioning, interrogation if you will, which we follow through with contact tracing."
Bolt said on social media Monday that he was awaiting the result of a coronavirus test and was quarantining himself as a precaution.
The 34-year-old retired sprinter who won gold in the 100 and 200 meters at the last three Olympics posted a video on social media to explain the situation.
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"Good morning everybody. Just waking up. Like everybody, checked social media and saw that social media says I'm confirmed of COVID-19," he said. "I did a test Saturday, because I work. I'm trying to be responsible, so I'm going to stay in and stay here for my friends."
Bolt, who appeared to be lying in bed in the video, said he has no symptoms.
"Just to be safe, I'll quarantine myself and just take it easy," he said.
Prime Minister Andrew Holness said police are investigating the case. He didn't specify what offense might have been committed, though Jamaica has imposed social distancing requirements due to the COVID-19 pandemic.
"They're looking into all aspects of the matter. So no one is going to be treated with any exemption or given any special treatment. All Jamaicans have a duty," he said.
Jamaica has had fewer than 1,700 cases, a relatively low number in a country of nearly 3 million.
On the track, Bolt was known for his poses - his "Lighting Bolt" became iconic - and of course his post-race celebrations. His speed and charisma made his sport can't-miss viewing whenever he lined up in the blocks.
Bolt set the world records in the 100 and 200 meters at the 2009 world championships in Berlin. He retired in 2017.