Galliani also unwittingly revealed potential transfer targets to the audience of radio station RDS.
Milan have slipped to 11th in the league, 23 points behind leaders Juventus and were ousted from the Italian Cup on Tuesday by 10-man Lazio.
Consequently, Galliani has become a target for unhappy fans and on Tuesday fell victim to the radio prankster, who pretended to be president of rival club Sampdoria, Massimo Ferrari.
The prankster quizzed Galliani if Inzaghi would be sacked if Milan lost to Lazio in Tuesday's last eight tie.
"No, unless there is a massive catastrophe we'll stick with him until the end of the season, then we'll see," said Galliani.
Galliani was also asked which strikers the club had targeted to shore up their ineffective attack.
"I'm looking for a centre-forward on loan and I will sign (Pablo) Osvaldo or (Mattia) Destro," said Galliani, before asking the fake Ferrero: "Are you planning to loan (Stefano) Okaka out? If so, then I'll be in touch."
In recent days Milan have been linked with Inter striker Osvaldo, who has been suspended after failing to turn up for training for several days, as well as Roma striker Destro.
Reports in La Gazzetta dello Sport on Wednesday said Milan had "agreed terms" with Roma to take Destro on loan until the end of the season for €700,000 with the option of a permanent move costing Milan a further 16m.
However, the report added that talks to take Italian striker Okaka from Sampdoria, who have recently signed Luis Muriel (Udinese) and Samuel Eto'o (Everton) were "ongoing".