The 42-year-old died at the team hotel in Paris where the Irish province were due to play Racing 92 in a European Champions Cup tie on Sunday.
The game was postponed as a mark of respect and tributes have been pouring in for the man capped 62 times by Ireland.
Ronan O'Gara said he was "heartbroken" by his former Munster and Ireland team-mate's death while a book of condolence has been opened and signed by hundreds at Munster's Thomond Park ground in Limerick.
Foley's family have been trying to come to terms with the devastating news and issued a statement outlining their gratitude for the rugby world's reaction.
"We wish to thank everyone for their support. It will help carry us through these darkest days," the statement read.
"With Anthony's passing, we have ultimately lost an amazing, adoring and loving father and husband; an equally caring, loyal and devoted son and brother; a central and go-to figure for the wider Foley and Hogan families.
"We know too that his sudden death has brought the rugby worlds of Shannon RFC, Munster, Ireland and much further afield crashing down.
"You have lost a former player, coach, friend and all-round inspiration - your and our hero both. We mourn his loss together."