The two players have agreed on a four-year deal with the Blaugrana, which was made official on Monday following months of speculation.
Kessie spent five seasons at San Siro, the first two of those on loan.
The Ivorian helped AC Milan to their first Scudetto triumph in 11 years in his final campaign with the club.
While Andreas Bødtker Christensen joined Barcelona from Chelsea
The player will sign a contract until 30 June 2026 and his buy-out clause will be set at 500 million euros.
The 26-year-old centre-back helped Chelsea win the Champions League in the 2020-21 season, the FIFA Club World Cup, and the Uefa Super Cup in 2021.
Christensen, who has 56 caps for Denmark, became a free agent after his contract with Chelsea expired last month.
Barcelona came second in LaLiga last season, finishing behind rivals Real Madrid.