The Premier League champions will part with £51m for the Norway international, who will sign on until the summer of 2027.
Haaland is one of Europe's most fearsome forwards, having scored 85 goals in 88 appearances since joining Dortmund from RB Salzburg in December 2019.
He has endured fitness problems this term but has still struck 28 goals in 29 games.
Haaland's father, Alf-Inge, also played for Manchester City between 2000 and 2003.
A statement on the club's website 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."
A statement from Dortmund added: "Erling Haaland is about to transfer from Borussia Dortmund to Manchester City. All parties have agreed on the basics of this today. The contractual details still have to be coordinated and documented.
"The implementation of the transfer is also subject to proper and timely processing in accordance with the provisions of the FIFA Transfer Matching System (FIFA TMS).
"With the realisation of this transfer business, the management expects a positive effect on the key earnings figures (EBITDA, EBIT) for the 2022/2023 financial year in the order of around EUR 35.0 million - 40.0 million.
"Due to the opening of the international FIFA transfer window from July 1st, 2022 and, among other things, other regulatory association requirements, the transfer business will not fall into the current 2021/2022 financial year, but into the 2022/2023 financial year."
Haaland's impending arrival at the Etihad Stadium will put to bed questions about Manchester City's need for an orthodox center forward following record goalscorer Sergio Agüero's departure last summer.
