Kompany suffered a fresh injury setback last week and before then, the Guardian reports he'd squared up to Kolarov at training.
The Serb has been impressive in place of Kompany at centre-half and the pair went "nose to nose" after a late challenge. Heated words were also exchanged before they were separated.
Kompany is facing another spell on the sidelines after breaking down in the EFL Cup win at Swansea City.
The Belgian is recovering from a groin injury.