France international Dembele scored 10 goals and provided 20 assists in all competitions to help Dortmund finish third in the Bundesliga last season. The team also won the German Cup.
A statement from Barca said Dembele would arrive in Barcelona on Sunday and have his medical on Monday before signing the contract, which will contain a release clause of 400 million euros. He will also be officially unveiled on Monday at the club's Nou Camp stadium.
"Without doubt he is (a) hugely talented player, one of the best on the continent," the statement on Barca's website added (www.fcbarcelona.com).
Shortly after Barcelona's announcement, a statement from the Bundesliga club confirmed the sale:
"FC Barcelona will pay Borussia Dortmund a fixed transfer compensation of 105 million euros. Above that there was an agreement of a variable transfer compensation up to a maximum of around 40 percent of the fixed transfer compensation."
Barcelona are also targeting Liverpool's Philippe Coutinho as they look to fill the void left by the departure of star forward Neymar, who joined Paris St Germain for a world record fee of 222 million euros.
After Neymar, Dembele becomes the second most expensive transfer of all-time as the deal's add-ons will take him past the 105 million euros Manchester United paid Juventus for Paul Pogba last year.
He also becomes the priciest player in La Liga, his fee outstripping the 100 million euros Real Madrid paid for Gareth Bale in 2013.
Barca's need to sign a top-class player to replace the Brazilian was underlined by their heavy 5-1 defeat to Real Madrid in the Spanish Super Cup earlier in the month, although the Catalans beat Real Betis 2-0 in their opening game of the Liga campaign.
"He's a player who guarantees an attacking threat, which is something we had lost and we badly need," Barca coach Ernesto Valverde told a news conference on Friday.
"He can play on either wing or through the middle, he gets forward a lot and is very technical. We expect a lot from him and we're very excited about him."
Asked whether Dembele was worth the 105-million-euro price tag, Valverde said: "We know what things are like in this market as the deadline approaches and everything becomes disproportionate.
"It seems like every club is going to sign the most expensive player of all time now because the prices are so high after Neymar's transfer. But when a player is on the pitch they don't worry about how much they cost."
Barca visit Alaves on Saturday, while Dembele is likely to make his debut after the international break on Sept. 9 when they host city rivals Espanyol.
(Reporting by Richard Martin in Barcelona, additional reporting by Aditi Prakash in Bengaluru, editing by Neil Robinson; Editing by Alison Williams and Christian Radnedge)
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