Biggar is likely miss the games against Scotland, England and Ireland, with Rhys Patchell, Owen Williams or Gareth Anscombe expected to wear the number 10 shirt.
Experienced Wales back-row forwards Sam Warburton, Taulupe Faletau and Dan Lydiate, and centre Jonathan Davies are also injured.
Wales start their Six Nations campaign against Scotland in Cardiff on Feb. 3 before meeting defending champions England at Twickenham.
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(Writing by Simon Jennings in Bengaluru, editing by Ed Osmond)
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