As first reported by tz and later by the Bild newspaper, the coach of the German record champions also dislocated his shoulder when he fell.
Kompany collapsed last Friday while watching the second team train at Bayern Munich's training ground on Säbener Straße. The cause remains unknown.
The 40-year-old was taken straight to hospital, where doctors are still treating and examining him. They have yet to reach a precise diagnosis, though Bayern have already given the all-clear on his condition.
"I visited Vince in hospital yesterday. He is in good spirits. It looks as though he is getting better every day. That is very positive," said assistant coach Aaron Danks, who is standing in for Kompany, at the press conference before the cup match against 1. FC Magdeburg.
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Why did Vincent Kompany collapse?
The report also says Kompany complained of severe headaches on the day he collapsed. He has since undergone extensive neurological examinations.
According to Bild, doctors are now focusing on a head-related problem as they try to reach a diagnosis. They quickly ruled out a stroke and a heart attack during the examination.
Kompany had also been suffering from a viral infection in the days before he collapsed. As a result, he missed the final training session and the press conference before the Champions League match against Arsenal (2-2), among other things.
How long the 40-year-old will be out remains unclear. According to Bild, he will not return for the matches against Dortmund (31.10.), away to Atletico Madrid (3.11.) or against Mainz (7.11.). Kompany may be back after the international break in November.