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Kompany collapsed while watching Bayern Munich’s reserve team train last week. The 40-year-old had been unwell with a virus and missed the pre-match press conference against Arsenal in the Champions League, although he was on the touchline during the game.
Sporting director Max Eberl issued an update on the head coach to German broadcaster ARD on Wednesday.
"Given the circumstances, he's doing well," said Eberl. "He was discharged from the hospital today. That is the biggest and best news for all of us."
No timeframe has been set for Kompany’s return to the dugout and the club said on Saturday it would be a matter of “weeks, not days”.
Assistant coach Aaron Danks has taken charge of the wins against Freiburg and Magdeburg in the head coach’s absence.
"It's a meningeal irritation that caused this incident on Friday. He will now rest and take medication," Eberl added. "I am very confident that he will be back on the touchline, operating at 100 per cent very soon."
Kompany spent two seasons in charge at Turf Moor, initially leading the Clarets to the Championship title and passing the 100-point milestone. His side was relegated from the Premier League the following campaign.
The 40-year-old left Burnley in the summer of 2024 to take charge of the German giants and has won back-to-back Bundesliga titles.