Silva sustained the knock in his side's 2-0 Capital One Cup defeat to Newcastle when tangling with Ryan Taylor on Wednesday.
"Manuel Pellegrini confirms that @21LVA will miss Sunday's game. Will be out for 3-4 weeks. #mcfc #cityvunited," City said on Twitter on Friday.
The Premier League champions will host United at the Etihad on Sunday, with kick-off at 15h30 South African time.