His funeral will take place on Thursday at the Basilica di Santa Croce in Florence.
Fiorentina called for respect and silence on Monday after Italy's football chief Giovanni Malago appeared to make a gaff by erroneously declaring a contract extension had been posthumously granted to the club's deceased captain Davide Astori.
A judicial inquiry had been launched following the death of the player, who was found dead in his hotel room in Udine, where his team was due to play Udinese on Sunday.
Astori made 289 Serie A appearances over 10 seasons and won 14 caps for the Italian national team.
In the press conference, Pochettino said (H/T the Daily Telegraph): "I want to send all my condolences to David Astori and all the people that love him, and I think it's a massive tragedy".
Tributes have continued to come in from across the football world, with Paris Saint-Germain's Dani Alves describing Astori as "an admirable person", while former AC Milan CEO Adriano Galliani said the former Rossoneri youth player was "one of our boys". "It's very hard at this moment to find the right words for the family", Chelsea coach Conte told Sky Sports ahead of his team's Premier League match at Manchester City on Sunday.
He made his worldwide debut under coach Cesare Prandelli in March 2011 as a first-half substitute in a 2-0 victory over Ukraine in Kiev before getting sent off with 17 minutes remaining.