The 24-year-old is the second starting signal-caller to hit the reserve list, as the Detroit Lions added Matthew Stafford on Saturday.
A 2019 sixth-round pick, Minshew took over the Jaguars' starting role last year following Nick Foles' early-season injury. He finished his rookie campaign with a 6-6 record while completing just over 60% of his pass attempts for 3,271 yards and 21 touchdowns against six interceptions.
The Washington State product is now set to be Jacksonville's Day 1 starter after the club traded Foles to the Chicago Bears this offseason.
The new reserve list is for any player who either tests positive for COVID-19 or is quarantining after coming in close contact with an infected person. Players who had contact with an infected individual may return to camp after three negative tests over a four-day span.