The actor, who is currently filming in Australia, posted a statement on Twitter saying: "We felt a bit tired, like we had colds, and some body aches.
"Rita had some chills that came and went. Slight fevers too. To play things right, as is needed in the world right now, we were tested for the Coronavirus, and were found to be positive."
Hanks, a two-time Oscar winner and star of films including Forrest Gump, Saving Private Ryan and The Post, is the highest profile Hollywood figure to be diagnosed with the COVID-19 virus.
The couple's son Chet later posted a video on Twitter saying that he had spoken to his parents and that they were doing fine.
Moments before Hanks posted on Instagram, Donald Trump addressed the US and announced a host of new measures, notably suspending all travel from Europe, excluding the UK, for 30 days starting from Friday.
The entertainment industry is also reeling from the outbreak and a host of major events have been cancelled.