Garay becomes the first La Liga player to publicly declare his status with Valencia confirming four more individuals associated with the club have also tested positive.
A statement released by Valencia on Sunday read: "Valencia CF inform that five positive cases of the COVID-19 coronavirus have been detected amongst first team staff and players," it read.
"All of these persons are now at home, in good health and obeying self-isolation measures.
"Besides providing further information over the coming hours, the club reiterate our support for the health authorities and their social awareness campaigns, and emphasise to all of the population the importance of staying at home and continuing to follow the previously published hygiene and illness-prevention measures.
"Likewise, Valencia CF reiterate our confidence in the Spanish health system, and in the recommendations made by the Ministry of Health for cases of coronavirus infection such as these.
Garay, who has endured a nightmarish start to the year after being ruled out for the rest of the season with a knee injury last month, posted on Instagram: "It's clear that 2020 has got off on the wrong foot."
"I have tested positive for the coronavirus; I feel very good and now I just have to listen to the health authorities.
"For the time being, I am isolated."
Spain is the fifth-worst affected Covid-19 country in the world after China, Italy, Iran and South Korea, having totaled more than 6,000 cases and 190 deaths.
The Spanish top flight has been suspended for at least the next two matchdays after some members of the Real Madrid basketball ball team, who share facilities with football players, tested positive for the virus.
The Spanish government is expected to put the entire country into lockdown on Monday in a bid to contain the spread of the virus.
