The Portugal international's future has been the subject of much speculation since it was confirmed that he was looking to leave the club last summer.
Silva ultimately stayed put at the Etihad to help Man City win their sixth Premier League title, and he remains under contract with the English champions until 2025.
Pep Guardiola admitted in 2021 that Silva was hoping to secure a move elsewhere, but Football Insider now claims that the opposite could occur as Man City try to sign him to a new deal.
The former Monaco man is reportedly viewed as an integral part of the club's future under Guardiola, who extended his own deal until 2025 earlier this month.
City are supposedly hopeful that Guardiola's new contract will sway players whose situations are more uncertain, and Silva's is now seen as the priority.
The 28-year-old is believed to earn as much as £150,000 per week at the Etihad under his current terms, which he signed in 2019, two years after making the switch from Monaco.
However, Silva can supposedly expect a pay rise with any fresh terms, and the overall value of the deal would be worth more than £40m.
Barcelona have long-been linked with a move for the Portugal international amid their own midfield uncertainties, with Frenkie de Jong and Franck Kessie both being tipped to leave in recent months.
There have been suggestions that Xavi's side are hoping to secure a deal for Silva in the January transfer window, with Gerard Pique's retirement helping them to free up the necessary funds for a deal.
Man City have ruled out the prospect of Silva leaving the club over the winter, though, and the new terms that City have offered him are believed to be for the long-term rather than a one or two-year extension.
Silva has now tallied 50 goals and 56 assists in 273 games for Guardiola's men in all competitions, winning a total of 11 trophies during his five years in sky blue.
This season, the ex-Benfica starlet has come up with two goals and five assists in 22 appearances, starting 12 Premier League matches for the reigning champions as they sit five points behind Arsenal in the table.
City managed to tie Phil Foden down to a new long-term contract in October, while Riyad Mahrez, Rodri, Scott Carson and Joao Cancelo have also penned extensions at the club in the current calendar year.