Norway striker Erling Haaland was born in Leeds in July 2000, the same summer his footballer father Alf-Inge left Leeds United to sign for Manchester City
Haaland, 21, will join in July after City activated a 60m euro (£51.2m) release clause in his contract.
The deal ends the club's hunt for a successor to their all-time top scorer Sergio Aguero, who joined Barcelona on a free transfer last summer.
"We can confirm that we have reached an agreement in principle for the transfer of striker Erling Haaland," City said.
"The transfer remains subject to the club finalising terms with the player."
Haaland, who passed his City medical in Belgium on Monday, has scored 85 goals in 88 appearances for Dortmund since joining from Red Bull Salzburg in January 2020.
La Liga giants Real Madrid and Barcelona also spoke to Haaland about a potential deal, but he opted for a move to Etihad Stadium.
Speaking earlier on Tuesday, Manchester City boss Pep Guardiola said he could not discuss the incoming signing for legal reasons.
"Everybody knows the situation. I shouldn't talk because I don't like to talk about the future or next season," he said.
When asked what impact a new number nine would have on the Premier League champions, Guardiola said: "New players always helps to be a stronger team. But not one defender or striker will solve our defensive or scoring problems.
"The club always has to have a vision for the future. The decisions the club makes I am aware and supportive of 100%."
Sports agent Mino Raiola had been involved in the discussions prior to his death on 30 April, with associates taking the process on to its conclusion.
Haaland's transfer will be one of the biggest of the summer, and he will score freely for City next season according to Jurgen Klopp, manager of their Premier League title rivals Liverpool.
"City will never be a team who win games because of one player," Klopp said.
"They have a specific way to play and I think Erling will realise all of a sudden that he will score a lot of goals on the second post if he just puts a foot on [the ball] and he will love that. We have other situations where he is a real beast.
"He was injured a couple of times at Dortmund but when he was fit, he was a real beast, so unfortunately [it is] a really good signing."
Even though City are within touching distance of a second successive Premier League title achieved without an orthodox centre-forward, Guardiola has always stressed his desire to have one.
City tried hard to get Harry Kane from Tottenham last summer but Spurs chairman Daniel Levy refused to sell.
And, with Kane having two years left on his contract, Haaland is an easier deal to conclude given his release clause.
He has scored 92 league goals in 121 appearances for Molde, Red Bull Salzburg and Dortmund and last year became the youngest player to reach 20 Champions League goals.
Haaland's father Alf-Inge played for City between 2000 and 2003 during a career where he also represented Leeds and Nottingham Forest.
-- Courtesy of BBC Sport