The Italy international has been linked with a move away from San Siro during the transfer window, with Roma, Borussia Dortmund and Liverpool all reportedly eyeing his signature.
El Shaarawy endured an injury-plagued 2013-14 season, but has made nine league starts under Inzaghi this term.
When asked about a possible transfer at a media conference, Inzaghi stepped in to take the question.
"I can answer for him: his future is with AC Milan," he is quoted as saying by The National. "Stephan is our flagship, one of our best players and his future is with us.
"He is trusted by us and by the president. He has been injured, but he is doing well. We all expect a lot from him. The new year will be the year of El Shaarawy."
El Shaarawy shares his coach's optimism, and added: "I have my personal goals, to come back and put in the best performances, and to score more goals.
"I have been lucky enough to wear the Milan jersey from the age of 18. It was always a dream for me, a dream come true.
"This is my fourth year at Milan and I have had some great times and troubled times. I hope to give a lot more to Milan and to Italian football."
