Schefter noted the team's Week 6 game against the Minnesota Vikings scheduled for Sunday is now in "jeopardy."
"Still working through details, can tell you we're not at four confirmed," an unnamed team official told Schefter, who'd reported there were four positive test results.
NFL Network's Tom Pelissero reported it was a Falcons assistant coach who tested positive and said the team plans to return to the facility tomorrow if there are no more issues.
The Falcons released a statement that confirmed one new positive test:
"Out of an abundance of caution following one new positive test, we have made the decision to stop all in-person work at IBM Performance Field Thursday and will conduct all operations virtually. This decision was made in consultation with the NFL and medical officials. The health and safety of our team is our highest priority."
The NFL recently dealt with its first widespread coronavirus outbreak involving the Tennessee Titans, who returned from a layoff that lasted over two weeks and caused several schedule changes to defeat the Buffalo Bills in a battle of undefeated teams Tuesday night.
In addition, the Kansas City Chiefs and New England Patriots have also dealt with smaller clusters of positive tests in recent weeks. Two of the Pats' most important players, quarterback Cam Newton and cornerback Stephon Gilmore, landed on the reserve/COVID-19 list.
The Falcons haven't come in contact with any of those teams during their 0-5 start.
Atlanta suffered a 23-16 loss to the Carolina Panthers last Sunday, and Carolina announced Wednesday it entered the NFL's more stringent "intensive COVID-19 protocol" after the Falcons placed defensive tackle Marlon Davidson on the reserve/COVID-19 list.
Carolina determined six of its players made direct contact with Davidson, leading to the change in protocol even though it hadn't recorded any new positive tests as of Wednesday, per Alaina Getzenberg of the Charlotte Observer. Three Panthers players spent time on the reserve/COVID-19 list earlier in the year.
The Falcons and Vikings both have their bye weeks still available, which makes a potential schedule change easier to navigate should Sunday's game be postponed. Bigger problems could arise for the NFL later in the year when teams have used their bye and thus can't move games around so freely.
Aside from a full postponement, games have also been delayed by a day or two to keep them within the same league week while awaiting a few rounds of negative test results before playing.