The Belgium international is set to undergo a scan on Monday, but the report claims City fear Kompany could miss as much as three weeks.
The 26-year-old centre-back will likely miss games against Queens Park Rangers, Liverpool and Southampton after limping off with calf problem during Saturday's FA Cup victory over Stoke City.
City assistant manager David Platt said: "It could be a fortnight, it could be three weeks. We'll know more this morning.
"My experience of calf strains would be no chance for tomorrow, Liverpool a big doubt with an outside chance, but that would depend on the gravity of it.
"In all honesty you're probably looking at the game after that being a potential realistic target."
City have informed the Belgian FA that Kompany will not be available for their friendly against Slovakia on February 6, and with Kolo Toure on Africa Cup of Nations duty, and Micah Richards and Maicon currently injured, City boss Roberto Mancini is short of options at the back.
Following the match, rumours circulated that the defending Premier League champions could dip into the market in an effort to strengthen in defence.
But Platt revealed that's not likely, with not much time left before the window closes.
"It's difficult to go into a market and pick people out just because you have an injury. We'll have to regroup and the manager will decide."
Matija Nastasic missed the trip to the Britannia Stadium at the weekend, as City progressed to the FA Cup fifth round with a late Pablo Zabaleta winner, but the Serbian international should be fit to face Liverpool on Sunday.
That means Joleon Lescott is likely to partner Nastasic at the back, as they look to keep in-form Liverpool striker Luis Suarez quiet.