Silva is a free agent after the end of the 2019/20 season and, according to Corriere dello Sport, is poised to sign a three-year deal with the Serie A side.
However, the deal will only be finalised once City’s season is over.
They face Lyon in the quarter-finals of the Champions League in Lisbon on Saturday, with the final taking place on August 23.
Silva is yet to win the Champions League and City manager Pep Guardiola said at the weekend: “David Silva deserves it more than anyone else.”
Silva signed for City in 2010 and has made over 430 appearances for the club.
VIEW FROM ITALY - A tactical dilemma?
"Not only the Corriere dello Sport, but various sources have already revealed that David Silva is very close to Lazio. We can confirm that the deal is expected to be completed soon," Eurosport Italy's Stefano Silvestri says.
"Silva should earn €4 million per season, less than his wages in Manchester.
"Lazio play in a 3-5-2 formation with a lot of quality in their midfield. They have Milinkovic-Savic and the former Liverpool duo Lucas Leiva & Luis Alberto. In my opinion, at the beginning Silva should compete with Milinkovic or Luis Alberto, without being a starter. It will be very interesting to see how Inzaghi will face this tactical dilemma."