Silva, who's leaving City after 10 years, is reportedly interested in playing in Italy next season.
Lazio face competition from Qatari side Al-Sadd, where Barcelona legend and Silva's former Spain teammate Xavi is manager. Xavi wants Silva to join him in the Middle East, according to Llorens.
Valencia were reportedly considering re-signing Silva before the coronavirus pandemic. The 34-year-old came through Valencia's academy and played four seasons with Los Che before heading to City in 2010.
Lazio managed to qualify for the Champions League group stage for the first time in 13 years, a detail that could persuade Silva to join them.
The World Cup winner will be available on a free transfer at the end of City's Champions League campaign. City currently hold a 2-1 aggregate lead over Real Madrid in the round of 16.
The Biancocelesti have seen success with veteran players. German striker Miroslav Klose scored 54 goals in 139 Serie A appearances and ended his career at Lazio in 2016. Former Liverpool midfielder Lucas Leiva also rejuvenated his career in the Italian capital.