Boateng has not played since he was released by Schalke in May for what the Bundesliga club said was a break down in trust between club and player.
"On January 4 we will register Boateng's contract with the Lega Serie A and he will be available for selection," Galliani said in an interview with an Italy's Sky TV after Milan's 4-2 away win at Frosinone on Sunday.
Galliani added that Spanish winger and former Liverpool player Suso will join Genoa on loan.
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(Reporting by Bartosz Dabrowski in Gdynia; editing by Justin Palmer)
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