The Belgium defender went off after 33 minutes of City's FA Cup tie at the Britannia Stadium and first-team coach David Platt confirmed he faced a spell out.
The news limits manager Roberto Mancini's defensive options at a time when Kolo Toure is on international duty at the African Nations Cup.
Platt said: "He has pulled his calf. We won't know how bad it is until the morning but there won't be any chance of Tuesday."
Remarkably it took City seven minutes to replace Kompany, who pulled up and headed straight down the tunnel, but City battled through to win 1-0 with a late Pablo Zabaleta goal.
Zabaleta struck from close range after 85 minutes as City, who had been dominant in the early stages, finally wore their resilient hosts down.
Platt said: "The pleasing thing is we battled hard, defended well and kept another clean sheet.
"It is pleasing from that perspective and it is also pleasing to go through.
"We knew it was going to be difficult here. You know the ball is going to come in the box and you're going to have to compete for first balls, second balls all day.
"It is the way it is going to be and I thought we did it well."
- Sports & Recreation
- David Platt
- Roberto Mancini
- Pablo Zabaleta