As of Tuesday morning, the team did not feel its home game Sunday vs. the Cincinnati Bengals was in jeopardy, a source told ESPN.
Ioannidis was placed on injured reserve in September with a torn biceps.
He is the first to test positive for Washington since the season began.
In a statement released Tuesday morning, the team said: "Last night we were made aware that a Washington player tested positive for COVID-19. The player immediately self-isolated and the contact tracing data was evaluated. All of the player's close contacts have been told to quarantine."
The statement also said that all meetings this week will be conducted virtually and the practice facility will be open to staff; players will have limited access. Washington is still trying to determine what it means for practices; there's a chance all of its practices will be walk-throughs this week.
Washington was the first team to pull its scouts off the road because of the virus early in the spring. It also announced Monday that, because of increasing coronavirus cases, no fans would be able to attend Sunday's game -- after allowing 3,200 fans to attend the previous home game.
The statement also said: "We will continue to work closely with the NFL's Chief Medical Officer and follow the league's guidance. The health and safety of our players, coaches and staff is our number one priority."