Mancini recently confirmed he was interested in El Shaarawy and Roma star Daniele De Rossi and could make his move for the pair in January.
The City manager has been mightily impressed with the AC Milan youngster this season, whose 13 goals in 16 Serie A games has led to interest from numerous clubs.
Milan are having an underwhelming season despite the Italy forward's bright start and there are concerns they could lose him if City offer big money.
There has even been talk of a swap deal involving fiery striker Mario Balotelli but Galliani has moved to reassure fans that El Shaarawy is going nowhere.
"El Shaarawy to swap with Balotelli at City? I swear that will end with El Shaarawy at Milan," Galliani told reporters in Italy following a charity dinner.
Balotelli was pictured posing with an AC Milan shirt when he played for city rivals Inter before he moved to Manchester and has once again been linked with an exit from the Etihad Stadium.