The 27-year-old sustained a treble fracture in one of his fingers during Bourgoin's 22-11 win over Clermont-Auvergne at the weekend and will undergo surgery on Thursday in Lyon.
Bonnaire started France's last three Test matches against Canada, Tonga and South Africa.
He should be back in action by mid-January and is likely to be available to France coach Bernard Laporte for the RBS 6 Nations campaign.
France, who finished second last season, will start the 2006
event on February 5 in Scotland.