Silva was due to leave the Etihad Stadium on June 30 and had held talks with former club Valencia about a return.
However, with games set to continue through the summer to make up for time lost because of the coronavirus pandemic, City have moved to secure the 34-year-old until the end of the campaign.
An official agreement will not be put in place until the plan for completing the season is made a official but sources have told ESPN that Silva is willing to stay, particularly given he still has the chance to win the Champions League -- the one major trophy missing from his collection.
FIFA have issued guidance that contracts due to end on June 30 should be extended until the season can be completed but both the club and the player must agree before it can he ratified.
City have 10 Premier League games remaining, plus outstanding fixtures in the FA Cup and Champions League. Pep Guardiola's side are 2-1 up in their Champions League round-of-16 tie against Real Madrid with the second leg at the Etihad still to play.
Silva has won 11 major honours during 10 years at the club, including four Premier League titles. He has made more than 400 appearances for City since his move from Valencia in 2010.