Norris threw a bullpen session on Tuesday and is scheduled to pitch a simulated game Thursday.
Norris, 27, said he was tested while still in Florida but didn't receive his positive result until he arrived in Detroit for the start of camp.
He said he was taking precautions while in Florida.
"It was crazy down there. Everything was like open and semi-normal, but I wasn't partaking in all that,'' he said. "I just kind of figured I was doing enough to not get it.''
Norris said his symptoms didn't seem major at first -- he woke up sweating a bit and achy -- but in retrospect, he realized those symptoms may have been more significant.
"The heavy symptoms only lasted a few days,'' Norris said. "A few days after that test, I was like, 'I feel like 100 percent.'''
He said he sent out a message to teammates telling them not to take the virus lightly.