The 34-year-old signed a two-year contract with Sociedad on Monday, having left City as a free agent at the end of the 2019-20 season.
The news provoked an angry response from Tare, as Stadio Olimpico had initially been touted as the most likely next destination for Silva following his Etihad Stadium exit.
"I have great respect for the player, but not for the man," the Lazio chief said of Silva's perceived snub.
The Spaniard's father has now come out to defend his son, insisting that the midfielder had not spoken to the Italian club directly and that a final deal had never been agreed.
Fernando Silva told Radio Marca: “Everyone can say what they want, Lazio spoke with their representative, not with David, so I don't understand why they name him as a person.
"He did not have a closed commitment with Lazio and there are other teams like Real that have also spoken with him.”
City announced plans to erect a statue of Silva outside of the Etihad after bidding farewell to the most successful player in the club's history.
The diminutive playmaker won 14 trophies during his 10-year spell in Manchester, including four Premier League titles, and recorded 77 goals and 140 assists for the Blues in 436 appearances across all competitions.
The former Spain international said his final goodbyes to City supporters via social media, writing on Twitter: "Manchester City and I have grown together. In 10 years, I have developed personally and professionally.
"I have celebrated many victories and titles and lived through epic moments.
"I am going to support the team from a distance. I am so happy I have shared these unforgettable moments with the City family - you will always be in my heart.
"I want to send every Cityzen a huge hug."