The AC Milan midfielder will move to Camp Nou on a free transfer, when his contract at the Serie A club expires on June 30.
Sources close to the talks told ESPN that Barca have sealed the signing of 25-year-old Kessie after agreeing the move's final details with the Ivory Coast international's agent.
Meanwhile, negotiations are progressing with Chelsea defender Christensen, also 25, who will be the next recruit to arrive when his contract runs out at Stamford Bridge.
ESPN reported earlier this month that both deals were on track, and that barring any unexpected developments, both would be Barcelona players.
Kessie, 25, has been a first-team regular at Milan since joining from Atalanta in 2017. He has scored five goals in 22 Serie A appearances this season after registering career-best figures, 13 goals in 37 league games, in 2020-21.
Resolving the issue of the commission due to Kessie's agent George Atangana was key to closing the deal.
Sources told ESPN that a solution was found in recent days after Atangana was registered with the Spanish football federation (RFEF) to be able to receive compensation for his role in the signing.
Centre-back Christensen will be the next incoming at Camp Nou, as Barca wait on developments in their bid to sign Chelsea teammate Cezar Azpilicueta on a free transfer.
Last November ESPN reported that Xavi had requested the signing of the versatile 32-year-old.
Azpilicueta is also out of contract at Stamford Bridge in June. He is keen to move to Camp Nou but the move has been delayed by the uncertainty around Chelsea following the UK government's decision to freeze the assets of owner Roman Abramovich.
Chelsea hold the option to unilaterally renew their captain's contract, but Barcelona believe the Premier League club may be willing to let him to leave for free out of gratitude for his successful 10-year spell there.
Meanwhile Barca are also working on a possible deal for Ajax full-back Noussair Mazraoui.
The 24-year old is another player whose contract expires this summer, and sources have told ESPN that talks are moving in the right direction.