Red Sox manager Ron Roenicke confirmed July 7 that Rodriguez had tested positive for COVID-19.
Roenicke told reporters Thursday the left-hander hadn't tested positive again but that Red Sox trainers "discovered something that we need to make sure we get rid of this completely before he gets on the field."
The team will re-evaluate Rodriguez in a week.
Bradford pointed to research that shows myocarditis could be one effect of the coronavirus for some patients. The Mayo Clinic writes that "a viral infection usually causes myocarditis," which is inflammation of the heart muscle. Chest pain, shortness of breath and fatigue are among the symptoms.
Roenicke described Rodriguez as "tired" after the pitcher threw around 20 pitches in a bullpen session.
Rodriguez's absence puts further strain on a rotation already without Chris Sale, who underwent Tommy John surgery this spring and will miss all of 2020.
Rodriguez made 34 starts for Boston in 2019 and finished 19-6 with a 3.81 ERA and a 3.86 FIP, per Baseball Reference. He tied for sixth in American League Cy Young voting with Chicago White Sox right-hander Lucas Giolito.
Boston opens its season Friday against the Baltimore Orioles.