Roenicke also announced Tuesday that prospect Bobby Dalbec has tested positive for the virus.
Rodriguez has yet to report to camp after previously informing the team that he had been around relatives who had been ill. Dalbec, a 25-year-old third baseman, is also at home and currently is asymptomatic.
Because of ace Chris Sale's season-ending Tommy John surgery and David Price's departure to the Los Angeles Dodgers, Rodriguez was expected to take a lead role in Boston's starting rotation this year. Roenicke said that he intended to have Rodriguez start on Opening Day.
"Just unfortunate that we might not be able to still do it," Roenicke said. "We'll have to wait and see."
Rodriguez, 27, enjoyed a breakthrough 2019 season and was one of the bright spots on the Red Sox staff, going 19-6 with a 3.81 ERA and 213 strikeouts. He posted career-bests in wins, ERA, strikeouts and innings (203.1).
The Associated Press contributed to this report.