The winger was withdrawn in the 66th minute of the UEFA Champions League clash, which saw Manuel Pellegrini's men progress to the knockout stages of the competition for the first time in their history.
Silva will now be absent for his side's trip to Sunderland on Sunday, and potentially for fixtures against Tottenham, Swansea City and West Brom.
He could also miss the Champions League clash with Viktoria Plzen.
"Obviously we have another day before we make any decisions, but David Silva, Richard Wright, Stevan Jovetic and Vincent Kompany will miss the Sunderland game," said Pellegrini.
"Fernandinho and Matija Nastasic have slight problems but could possibly be fit for Sunday.
"As I say, we have some more time but we will have to wait and see. Silva has a problem with his calf so he could miss three or four weeks."
Pellegrini is hopeful that captain Kompany could return to full fitness after the international break.
