The CSKA Moscow attacking midfielder, who becomes a free agent at the end of this month, will become Milan's first signing in the winter transfer market.
"Now we can say it," Galliani told the club's official website, www.acmilan.com. "From January 3, 2014, Keisuke Honda will be a Milan player."
Honda joined the Russian giants in 2010 from Eredivisie side VVV-Venlo and has developed into one of the game's most sought-after talents.
The 27-year-old could make his Serie A debut against Atalanta at the San Siro on January 6.