Galliani, the Rossoneri's former chief executive, said he had spoken with the Swede in an audacious attempt to lure him to the Serie C side.
Confirmation of the contract offer comes after former Milan owner Berlusconi, who purchased Monza in 2018, publicly outlined his desire to reunite with Ibrahimovic.
The 38-year-old has been a free agent since leaving LA Galaxy after two seasons in MLS.
"After Berlusconi gave me a mandate, I made the offer to [Ibrahimovic's agent] Mino Raiola," Galliani, now CEO at Monza, told Corriere della Sera.
"He suggested I talk about it directly with Zlatan, who I speak with every night."
Galliani explained his pitch: "Come to us, stay six months in Serie C. Then next year we go up to Serie B and in two seasons we are in Serie A."
Monza is on track to win promotion this term, but signing Ibrahimovic would still represent a major surprise.
The former Barcelona, Paris Saint-Germain and Manchester United star has been heavily linked with either a return to Milan or a move to Napoli.
Rumoured suitors Bologna have ruled themselves out of the race, with director Walter Sabatini telling Tuttomercatoweb: "Ibrahimovic will not be coming to Bologna. He has made other choices."