Milan have overhauled their side with the signings of Leonardo Bonucci, Lucas Biglia, Andre Silva, Hakan Calhanoglu, Ricardo Rodriguez, Mateo Musacchio, Andrea Conti, Franck Kessie and Fabio Borini, but the Serie A giants are still keen on another striker.
Andrea Belotti and Pierre-Emerick Aubameyang have all been linked but Fiorentina's Kalinic appears set to move to San Siro.
"There's no point denying we have been negotiating for Kalinic for a long time. We are in advanced talks," Fassone told reporters following Thursday's 6-0 rout of Shkendija in the first leg of their Europa League play-off tie.
Former striker Ibrahimovic has also been linked with a return to Milan, after his contract was not renewed by Manchester United.
Milan have previously refused to rule out a move for the 35-year-old, who scored 28 goals in all competitions at United last season, and Fassone reiterated that stance after Ibrahimovic's agent Mino Raiola – also a representative for Gianluigi Donnarumma – was spotted at the stadium.
"I don't know, he had a meeting with [director Massimiliano] Mirabelli, not with me. He will have talked about his clients and their future," Fassone said when asked about Raiola.
"Will there be another gift? The fans are happy with what we are doing. We've had a chance over the last few days to consider what is still missing and we're seeking it without haste.
"If the right opportunity comes up, then we'll take it, otherwise we are competitive as we are."