The defender, who was thanks to miss the remainder of the season after undergoing surgery on a ruptured anterior cruciate ligament last month, made the announcement on Instagram on Sunday but said he feels "very well".
LaLiga is currently at the beginning of a two-matchweek break while the Spanish government is about to declare a 15-day national lockdown on Monday in an effort to fight the pandemic.
Garay wrote: "It is obvious I even have started 2020 badly.
"I have tested positive for coronavirus but feel alright . Now it only remains to concentrate to the health authorities and, for now, be isolated."
Garay, 33, started 17 LaLiga games for Valencia this season and 4 within the Champions League.
Albert Celades' team are seventh within the Spanish top-flight, four points adrift of fourth-placed Real Sociedad. Their Champions League adventure came to an end last week with an 8-4 aggregate defeat to Atalanta.
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