Kimmich, who has not yet been vaccinated against the virus, has not played for Bayern since 7 November.
The 26-year-old was considered a close contact of Bayern team-mate Niklas Sule, who returned a positive PCR test result while on international duty with Germany last month.
After going through a period of isolation, Kimmich was forced to quarantine again when he was classed as a contact of another suspected COVID-19 case. He then tested positive for the virus himself later in November.
Kimmich has stated he is unwilling to receive a coronavirus vaccine until more is known about possible long-term effects.
Although Kimmich is no longer suffering from COVID-19, Bayern has confirmed he will not return to action until January.
Kimmich explained this was due to "slight infiltrations" in his lungs, which mean he has been unable to train fully.
"I'm happy my self-isolation caused by the coronavirus has ended," Kimmich said in a statement released on Bayern's official website.
"I'm doing very well, but I'm not yet able to train fully due to slight infiltrations in my lungs. I'll therefore do some rehabilitative training and can't wait to be fully back in action in January."
Bayern, which is having to play its home games behind closed doors due to coronavirus restrictions imposed in Bavaria, faces Mainz, Stuttgart and Wolfsburg before the Bundesliga takes its winter break.
Julian Nagelsmann's side beat Barcelona 3-0 on Thursday (AEDT) in the Champions League, a victory that followed up Sunday's contentious 3-2 win over Borussia Dortmund.
That Klassiker success lifted Bayern four points clear at the top of the Bundesliga.