The 32-year-old started the 34-7 French Top 14 win over Brive but was forced off after just 10 minutes due to the injury -- to be replaced by compatriot Blair Connor.
Ashley-Cooper, who was not picked by Australia coach Michael Cheika for the June Test series against England, is expected to miss the next three weeks.
With only one round of the Top 14 regular season left and Bordeaux far from sure of making the play-offs, Ashley-Cooper's season could be over.