Silva was hurt in the early stages of Wednesday night's 2-0 home defeat to Newcastle in the Capital One Cup as he tangled awkwardly with Ryan Taylor.
"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 injury means Silva will also miss next week's crucial Champions League clash against CSKA Moscow and a trip to QPR
He could also be unavailable for Spain's upcoming Euro 2016 qualifier against Belarus and a friendly with Germany.
Pellegrini, speaking at his press conference to preview Sunday's game, said: "David has a problem in his ligament
Maybe he will be out around three weeks so he will not be available for the derby.
"He is a very important player but I think we have a squad and we have to replace him with another player."
Midfielder Yaya Toure was also withdrawn as a precaution during the Newcastle defeat due to a groin problem but Pellegrini expects him to be fit to feature this weekend.
Pellegrini said: "Yaya has a small problem
We will see him on Saturday but I think he will not have any problems."
Also on the positive side, playmaker Samir Nasri appears to have come through his return to action in midweek, after seven games out following groin surgery, unscathed.
Source : PA
Source: PA