The Norway forward, 21, joins the Premier League leaders after a release clause in his contract was triggered, reported to be just £51million instead of the £63million that had previously been claimed.
A statement read: "Manchester City can confirm we have reached an agreement in principle with Borussia Dortmund for the transfer of striker Erling Haaland to the Club on 1st July 2022.
"The transfer remains subject to the Club finalising terms with the player."
It ends years of speculation over Haaland's future, with City fighting off competition from some of Europe's biggest clubs to secure arguably the hottest young talent in world football.
The former Molde and RB Salzburg striker scored 85 goals in 88 appearances for Dortmund in all competitions after signing for the Bundesliga club in 2020.
Manchester United tried and failed to sign Haaland in that transfer window, while the Norwegian has also been linked to Chelsea, Real Madrid and Barcelona in recent years.
City have played without a recognised centre-forward for most of the 2021/22 campaign after failing to sign Tottenham striker Harry Kane last summer.
Nevertheless, Pep Guardiola's side are the Premier League's top scorers this season and currently hold a slim advantage over Liverpool in the title race.
The Champions League remains elusive for City after their shock defeat to Real Madrid in the semi-finals - and Haaland's record in the competition is astonishing, with 23 goals in 20 games.
Legendary Rangers striker Ally McCoist believes the deal, widely reported to be £63million until the day of the confirmation, is a 'bargain' and expects Haaland to make City 'double unstoppable.
The former Golden Shoe winner told talkSPORT: "He's one of the best centre-forwards in the world and I can only see him getting better.
"I can't believe what I'm about to say, from a man who still remembers the first £100,000 transfer and Trevor Francis, the first £1m transfer.
"I'm about to say it's a brilliant bit of business. Clearly, his wages on top will make it a monster deal.
"But £63m is probably 50 per cent of what you would normal pay, probably more.
"I'm categorically astounding my self by saying City could be getting a bargain."
McCoist added: "He could potentially be the final piece in that jigsaw.
"If they get him firing, they could be double unstoppable. Of course they're brilliant, easy on the eye, and they've scored a lot of goals with the 'false no. 9'.
"However, if you've got the most exciting centre-forward in world football, in terms of potential, it's going to make some difference.
"The potential and the power and the strength of Haaland will make City a far better team.
"We know they want the Champions League, it was heartbreaking in Madrid, but if they get Haaland, dear me, can anybody stop them?"