The Citation CJ2 came to a stop on the grass at Norwich Airport just after 18:00 GMT on Saturday.
No-one was injured in the incident, an airport spokeswoman said.
The defender, who did not play in his team's FA Cup tie against Norwich, is understood to have been among five people on board.
The airport was closed to "fixed wing traffic" while the aircraft was removed, the spokeswoman said.
"The airport remained open to helicopter traffic. The Air Accident Investigation Branch has been informed, which is standard practice," the airport announced in a statement.
Kompany is currently sidelined with a calf injury. His team beat the Canaries 3-0 in Saturday's third round tie at Carrow Road.
A spokesman for the plane's operator, Centreline Air Charter, said it would not be making a comment until investigations had been completed.
Manchester City captain Vincent Kompany was on a charter plane which came off the runway during take-off, the BBC understands.