The club did not specify whether the individual was a player or member of the organization's staff.
"A member of the Denver Nuggets organization tested positive for COVID-19," the team statement said.
"The person, who was tested after experiencing symptoms consistent with COVID-19 on March 16th, is currently under the care of team medical staff and in self-isolation.
"The testing was undertaken following guidance from state public health officials and team physicians.
"The health and safety of our players, our organization, those throughout our league, and all those potentially impacted by this situation is paramount."
A number of NBA players have tested positve for COVID-19 since Utah's Rudy Gobert was confirmed to have contracted the disease last week, prompting the league to impose an indefinite suspension on play.