Having been diagnosed with lung cancer earlier this year, reports in Australia indicated Greig suffered a heart attack at his home in Sydney and died around 1345 AEDT (0245 GMT).
"He was rushed into St Vincent's hospital. The staff of the emergency department worked on Mr Greig to no avail," St Vincent's spokesman David Faktor was quoted as saying in the Sydney Morning Herald.
Australia captain Michael Clarke was among the first to express his sadness at Greig's death, and he said on www.cricket.com.au: "I was only speaking with Tony a couple of days ago so news of his passing is absolutely devastating.
"Tony has a long and decorated history with international cricket both as a player and commentator and cricket will be much poorer for his loss.
"Personally, he has also been a great mentor for me, providing great advice through the good times and the bad. The news will hit the cricket community hard, but we will never forget the lasting legacy Tony leaves us with.
"On behalf of the Australian cricket team our thoughts, prayers and wishes are with (the) Greig family at this difficult time."
Greig first became aware he had a problem in Dubai during Australia's one-day series against Pakistan, on which he was commentating, in August and September.
Initially diagnosed with bronchitis in May, the condition lingered and, by the time of the ICC World Twenty20 - which finished in Sri Lanka in October - tests that revealed a small lesion at the base of his right lung. On his return to Australia, he had fluid removed from the right lung. Testing revealed he had lung cancer.
CA chairman Wally Edwards said: "On behalf of Australian cricket, I offer condolences to Tony's family, friends and fans and admirers."