"The Beckham deal started with him," Galliani told Italy's state RAI TV. "It was the English player who had his lawyers tell us to call, after expressing an intention to play for Milan."
Galliani announced last week that Milan was close to signing the former England captain on a loan deal from the Major League Soccer's Los Angeles Galaxy.
The 33-year-old Beckham is keen to play in Europe in the MLS offseason to keep match-fit and available for England coach Fabio Capello.
"Beckham will very probably come our way at the beginning of January and stay with us for two or three months, then go back," Galliani said. "It's not assured that he'll come."
At Milan, Beckham would join a stellar midfield with Andrea Pirlo, Gennaro Gattuso, Clarence Seedorf and Massimo Ambrosini.