Kessie, 25, has joined Barca on a four-year deal after his contract expired with Milan on June 30.
The midfielder has a €500 million release clause in his new contract, which runs to 2026. Barca will present him on Wednesday.
The Ivory Coast international's move to Camp Nou ends his five-year stay at the Rossoneri where he made 224 appearances and scored 39 goals.
He played a central role this season as Milan won a first Serie A title in 11 years, scoring six times in 31 appearances, including one against Sassuolo last weekend as the league was wrapped up.
Meanwhile, Denmark international Christensen, 26, joins Barca after his contract expired with Chelsea last month.
The defender's move to Camp Nou brings an end to his 10-year stay at Chelsea, where he made 161 first-team appearances.
After making his senior debut for Chelsea in 2014, Christensen completed a two-year loan spell at Bundesliga side Borussia Monchengladbach between 2015 and 2017.
He has featured regularly under several different coaches since returning to Stamford Bridge, helping the club win the Europa League in 2019 and the Champions League in 2021.