Klose picked up a torn thigh muscle while on international duty for Germany in Kazakhstan on Tuesday and will be out of action for at least 10 days.
"He will not play against Hanover and he will not be available for the Champions League match against Cluj (next week)", the club said in a statement on Thursday.
Bayern, 12th in the standings and 13 points behind leaders Mainz after seven games, will also be without Bastian Schweinsteiger and captain Mark van Bommel, whose aggravated an injury on international duty for the Netherlands this week.
Striker Ivica Olic is another player doubtful for the match against third-placed Hanover after breaking his nose in Croatia's qualifier against Israel on Oct. 9 and undergoing surgery on Tuesday.
"Further tests will determine whether he can play with a protective mask," the club said.
GOMEZ SCORES
Bayern have only scored five goals in the league after losing Arjen Robben and Franck Ribery to long-term injuries early in the season and with his preferred strikeforce unavailable, coach Louis van Gaal will be forced to turn to Mario Gomez.
The Bundesliga's most expensive signing at around $30 million has yet to deliver on his inflated fee since joining Bayern last year and has spent most of his time on the bench.
Gomez, however, was on target in Germany's 3-0 against Kazakhstan when he came on as a substitute for Klose.
"The goal I scored was a step in the right direction and the confirmation of my hard work in the past weeks. Now I hope I can also score against Hanover," Gomez told reporters.
Surprise leaders Mainz, who have won all seven of their games with the most potent attack in the league, will break the record for the best start to a Bundesliga season if they beat visitors Hamburg SV on Saturday.
Second-place Borussia Dortmund, three points off Mainz, are looking to extend their own six-match winning streak when they travel to troubled Cologne, who sit three spots off the bottom in 16th.
In an clash of teams who were expected to be near the top of the standings, last season's runners up Schalke 04 (17th) host bottom club VfB Stuttgart, who sacked coach Christian Gross on Wednesday after picking up just three points from seven games.
