Peaty could compete in either the men’s or mixed 4x100m medley relay competitions in Paris, and was part of the squad that won gold in the mixed event in Tokyo.
Their title defence begins on Friday morning in the heats with a potential final, should Britain qualify, on Saturday evening.
The men’s medley relay begins on Saturday morning, with the heats on Sunday night.
Following his positive Covid test, Peaty said: “I’ll now be focusing on a fast, full recovery to give my best in the team relays later in the week.
“I’ve had so many messages and I’ll get back to you all, thank you for your support as it has truly been an unforgettable journey.”
The Team GB coaches would pick Peaty for whichever event they viewed as the best chance of a medal, possibly both if he felt fit enough, and he would be crucial to their hopes.
There are other breaststrokers in the GB squad who could take his place if required.
Peaty was targeting a third consecutive 100m Olympic title on Sunday, which would have meant he joined Michael Phelps as the only man to have won the same Olympic swimming event three times in a row.
He was well placed in the final 25m but Italian Nicolo Martinenghi came through to win gold.
Peaty gave emotional interviews afterwards when he discussed his difficulties since his last gold in Tokyo, which included problems with alcohol and his mental health.
He almost walked away from the sport before returning last year.
The Englishman is not the first aquatics athlete at these Games to test positive for Covid-19.
Five members of Australia’s water polo squad, which takes place at a different venue to the swimming, contracted the disease days before the Games.