Bayern sporting director Max Eberl delivered the positive update ahead of the defending German champions' DFB-Pokal match against 1. FC Magdeburg - which they won 6-1 on Wednesday night.
"I think this is the biggest and best news for all of us," Eberl told ARD.
He explained that doctors had identified the cause of Kompany's collapse, which occurred on the sidelines of a training session involving Bayern's second team last Friday.
"They found out what it is: meningeal irritation caused the incident last Friday," Eberl said.
A viral infection had previously been mentioned as a possible explanation for the incident.
Kompany has now been allowed to return home, where he will continue his recovery with rest and medication.
"But he has now been allowed to go home and can complete his rehabilitation there," Eberl said.
"For him, rehabilitation means rest and medication. And then I am very, very sure that he will soon be back on the touchline at 100 per cent."
However, Bayern will not rush the Belgian coach back into work, with his health remaining the priority.
Eberl stressed that Kompany would be given sufficient time to recover fully before returning to the touchline.
"He will be given all the time he needs - we have always said that," the sporting director added. "That is why we spoke of weeks from the very beginning."
Bayern Munich assistant coach Aaron Danks insists there will be no major changes to the team's preparations for Saturday's Bundesliga showdown against Borussia Dortmund, despite the continued absence of head coach Vincent Kompany.
Joshua Kimmich could make his return to Bayern Munich's starting line-up for Saturday's Bundesliga Klassiker against Borussia Dortmund after resuming training following a family bereavement.
Borussia Dortmund coach Niko Kovac has challenged his players to produce their best performance of the season when they travel to face Bundesliga leaders Bayern Munich on Saturday.
Borussia Dortmund coach Niko Kovac has not ruled out Konstantinos Karetsas playing in Saturday's Bundesliga Klassiker against Bayern Munich, with the attacking midfielder battling to recover from an ankle injury.
