Lovullo added Calhoun is asymptomatic and feeling great.
Calhoun is the third player on the D-Backs' 40-man roster to test positive for the virus.
Under MLB guidelines, the 32-year-old will be placed into quarantine and must return two negative tests before being allowed to participate at the club's summer camp.
Calhoun signed a two-year contract with the D-Backs in December after spending his first eight big-league seasons with the Los Angeles Angels.
He hit .232/.325/.467 with 33 home runs, 29 doubles, and 74 RBIs in 2019.