The Colts remain scheduled to face the Cincinnati Bengals on Sunday.
Indianapolis shut down its facility in the morning and began working remotely after discovering the four initial positives. The false results came from one player and three staffers, sources told ESPN's Adam Schefter.
The NFL has encountered several COVID-19 scares in recent weeks, with the most notable situation being an outbreak on the Tennessee Titans.
The Colts enter Week 6 with a 3-2 record, good for second place in the AFC South.