Kessie joins from Italian champions AC Milan while Christensen arrives from Chelsea. Both have agreed four-year deals with Barcelona.
The 25-year-old Kessie spent the past five seasons in Milan after first arriving in Italy with Atalanta in 2015.
He scored seven times in 39 appearances for Milan last season as the club won the Serie A title for the first time since 2011.
Christensen, 26, made 161 Chelsea appearances over 10 years at Stamford Bridge, winning a number of trophies including the Champions League and Europa League.
He had not played for the Blues since withdrawing himself from selection on the morning of the FA Cup final against Liverpool, which Chelsea lost on penalties.
Kessie will be presented by Barca on Wednesday, with Christensen set to be unveiled on Thursday.