The defender, who also played 14 times for Italy's national team, was discovered by Fiorentina staff when he failed to show up for breakfast at the hotel in Udine.
Speaking ahead of Chelsea's clash with Manchester City on Sunday, Conte paid a special tribute to Astori. "Congratulations really, you were one of the best sports figures I came across", the 40-year-old legendary goalkeeper posted on Facebook.
Udine state prosecutor Antonio Di Nicolo was quoted by the ANSA news agency as saying it was believed "that the player died from a cardiac arrest, the result of natural causes".
'I had him with the national team, i repeat it's very hard to speak in this moment about this tragedy.
Radja Nainggolan, Astori's former Cagliari teammate wrote on Instagram: "A great player, but even more than that, a great person.
The body has been taken for an autopsy, which I think will be conducted today".
Astori also earned 14 caps, or worldwide match appearances, for the Italian national team. His debut in March 2011 in a 2-0 victory over Ukraine in Kiev was eventful - he was introduced as a first-half substitute before being sent off with 15 minutes remaining after receiving two yellow cards.
Astori scored his only worldwide goal in the 2013 Confederations Cup third-place play-off against Uruguay, giving Italy a first-half lead in a match which finished 2-2.
The other five Serie A matches scheduled for Sunday, including the Milan derby, were also cancelled as a mark of respect.
"For this awful and delicate situation, and above all out of respect for his family, we appeal to the sensitivity of everyone", the team said in a statement.
The players in Genoa were warming up on the pitch when they were told of Astori's death and several walked off in tears.
"A great player, but an even bigger man, so many battles fought together in Cagliari before coming to Rome, I can not believe it", he wrote.
"A man who loved football and who grew up as a footballer with us. Goodbye captain", said former Italian prime minister Matteo Renzi. I don't believe it and I cry with the family and everyone at Fiorentina.