The former Barcelona boss underlined the importance of family, saying that Silva was free to miss future City matches to deal with his son's health issues.
"I want to thank all of you for the love and good wishes received in the last few weeks," Silva said in a statement on his official Twitter account.
"Special thanks to my teammates, the manager and all at the club for understanding my situation.
"Also I want to share with you the birth of my son Mateo, who was born extremely preterm and is fighting day by day with the help of the medical team."
Silva has been a mainstay in City's squad since joining in 2010 and has scored five goals in 19 league appearances for the league leaders this campaign.
City host Burnley in the FA Cup third round on Saturday, followed by the first leg of their League Cup semi-final against Bristol City on Tuesday.
(Reporting by Aditi Prakash in Bengaluru; Editing by Christian Radnedge)
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