"Juventus is currently activating all the isolation procedures required by law, including those who have had contact with him," the team added.
The Italian international is the first player in one of Europe's top leagues to contract the COVID-19 virus.
"I want to reassure all those who are worried about me, I'm fine," Rugani said on Instagram.
"At this moment, however, I feel even more duty to thank all the doctors and nurses who are struggling in hospitals to cope with this crisis. I invite everyone to respect the rules, because this virus makes no distinctions! Let's do it for ourselves, for our loved ones and for those around us."
All sporting events in Italy, including Serie A matches, are suspended until April 3, with the country's 60 million people under lockdown as the government attempts to combat the rapid spread of the coronavirus. Italy, the worst-affected country in Europe, has more than 12,000 confirmed cases, with a death toll of 827.
Prime minister Giuseppe Conte announced on Wednesday that all shops in the country, except grocery stores and pharmacies, would be closed. Earlier in the day, the World Health Organization (WHO) officially deemed the coronavirus outbreak a "pandemic."
Rugani was an unused substitute in Juventus' most recent match, a 2-0 victory over Inter Milan in Turin. Photos after the match showed him celebrating in the locker room, in close proximity with his teammates.
In the wake of Juventus' announcement about Rugani, Inter confirmed that they are suspending all competitive activities "until further notice."
Juventus, who last month quarantined their entire Under-23 side after an opponent tested positive for the coronavirus, are scheduled to play the second leg of their Champions League last-16 tie against Lyon next week behind closed doors at the Allianz Stadium in Turin.