Kayser, a regular for France, sustained a partial ligament rupture in his right knee during a French Top 14 match on Friday.
"He will be sidelined for up to six weeks," his club side Clermont said on their website (www.asm-rugby.com).
France, level with Ireland on four points at the top of the table, play their third game of the Six Nations season in Wales on Friday.
The 27-year-old Mach, who was diagnosed with thyroid cancer in 2012, seems likely to make his international debut against Wales.
(Reporting by Gregory Blachier; editing by Tony Jimenez)
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