The 31-year-old had only returned at the end of August after fracturing the same foot in April.
He suffered the fresh injury in training on Monday as Bayern prepared for Tuesday's German league match at Schalke.
"The fact that Manuel Neuer has suffered a new injury means we are all very sorry for him", said Bayern chairman Karl-Heinz Rummenigge.
"The operation went optimally and that is the most important thing now.
"Manuel will be available for us again, back at his old strength, in January."
Neuer's latest injury blow sparks concerns about Germany's skipper nine months before the World Cup kicks off.
He last played for Germany in a World Cup qualifier in October 2016.
Neuer will miss Germany's key World Cup qualifier against Northern Ireland in Belfast on October 5, where a win will book the holders' place at Russia 2018.
With Neuer out, Barcelona's goalkeeper Marc-Andre ter Stegen, who deputised when Germany won the Confederations Cup in July, is set to fill in for the world champions.
Neuer will miss Bayern's remaining Champions League group stage games, including their key away match at Paris Saint- Germain on September 27.
He will also miss their league and cup double-header against RB Leipzig at the end of October and away Bundesliga match at current leaders Borussia Dortmund in early November.
Neuer's Bayern and Germany team-mates quickly rallied around.
"You will come back stronger, my friend," wrote Bayern winger Franck Ribery on Twitter, while Thomas Mueller, Jerome Boateng and Robert Lewandowski offered a 'get well soon'.
Bayern confirmed Neuer underwent the operation at a clinic in Tuebingen, south-west Germany, by head surgeon Ulrich Stoeckle, who has inserted a plate in his foot.
Stoeckle has previously treated stars Bastian Schweinsteiger, Franck Ribery and David Alaba.
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