Elliott, speaking on a Twitch feed, said he had shortness of breath and a cough for a couple of days.
"Now, I'd say I feel good," he said.
Elliott, 24, said he believed that if he were tested again, the result would come back negative. However, he said he decided to give it a few more days.
The three-time Pro Bowler was perhaps the most high-profile NFL player to reveal that he had the coronavirus. As cases of positive tests for the virus mount, he said he was concerned about the safety of players, with the NFL preparing for a return to training camps in July amid the pandemic.
"I just feel like there's a lot of moving parts that have to be figured out," Elliott said. "I just don't know how they can keep the players [healthy]. You gotta put the health of the players first.
"And it's not even so much, I would say, the players' health -- because I got corona, and it really didn't affect me much. But a lot of people have kids -- they may have kids with asthma; they may have newborn babies; their parents or grandparents may live with them. ... We have to find ways to make sure that players and their families -- and the coaches also and their families -- aren't put at risk."
Texas, like Florida and Arizona, has seen a sharp spike in positive tests for COVID-19, with Gov. Greg Abbott saying Wednesday that the state is in the midst of a "massive outbreak."
Against that backdrop, Elliott made it clear that the safety of the players and staff and their families is his primary concern.
"Hey, I hope we play," he said. "I wanna let you guys know: I do hope that we have a season. But it has to be right. It has to -- we gotta put the health of the players and the coaches and the support staff first. I think that's important."