The midfielder will not be involved in Sofia when his country play their latest Nations League match on Thursday (AEDT).
Ramsey captained Wales in its 0-0 draw with Republic of Ireland on Monday, playing the full 90 minutes, but has been ruled out against Bulgaria with an unspecified injury.
He had previously missed the friendly defeat to England after a coronavirus outbreak at Juve.
The problem for Ramsey, who has started both of Juve's Serie A matches so far this season, was announced by the Football Association of Wales (FAW) on Tuesday.
That was also when the Portuguese Football Federation (FPF) confirmed his club team-mate Cristiano Ronaldo has tested positive for coronavirus.
Ronaldo is asymptomatic and "doing well", the federation said in a statement, and the 35-year-old has left the national team camp to begin self-isolating.
Juve also has a concern over the fitness of Paulo Dybala, who has been ruled out of Argentina's match with Bolivia due to illness, having also missed their World Cup qualifying win over Ecuador last week.
The Italian champion returns to action with an away game against Crotone on Sunday.