The 42-year-old, who won 62 caps for Ireland, was in France to prepare Munster for their European Champions Cup clash against Racing 92.
That match, scheduled for Sunday, has been postponed.
A joint statement from the IRFU and Pro12 outfit Munster did not disclose any further details of Foley's passing.
"Munster Rugby management are liaising directly with Anthony's family and will provide them with any assistance and support required," it added.
"The IRFU and Munster Rugby pass on our deepest sympathies to all of Anthony's family and friends and ask for privacy for the family at this sad time."