After City's elimination by Lyon on Saturday night, Silva was officially free to sign for a new club and Lazio appeared to be in pole position to secure his signature.
But it was announced on Monday night that Silva had instead opted to join La Real.
Sky Italia says this has angered Lazio.
It is claimed Silva didn't tell the Italian side that he was going to reject their offer and they only discovered he had moved elsewhere when Sociedad announced it. Lazio are said to be "furious" at the snub.
A move to Lazio appeared to be all wrapped up with Silva reportedly set to earn €4m (£3.6m) per season, as well has having access to a private jet and other luxurious perks.