Peaty shared the silver with American Nic Fink after both swimmers finished 0.02 seconds behind Italy's Nicolo Martinenghi.
The Briton had said he woke up unwell on the day of the race, although he declined to use that as an excuse. Peaty also struggled to speak after the race with a hoarse throat.
"In the hours after the final, his symptoms became worse and he was tested for COVID early on Monday morning. He tested positive at that point," Team GB said in a statement.
"He is hopeful to be back in competition for the relay events later in the swimming programme.
"As in any case of illness, the situation is being managed appropriately, with all usual precautions being taken to keep the wider delegation healthy."
The relay events will take place later this week.
Peaty said he had woken up to a 'curveball' on Sunday and had been trying to nurse himself back to full health.
"It's not an excuse at all, because I don't ever want that to be an excuse. But it's a curveball that I've had to respond to," Peaty said.
"And again, you can train eight years for something and not feel 100 percent on the day. And 100% costs you 0.02 (seconds), and that's just the way it is."
