Haaland is among the hottest transfer prospects this summer with the Norwegian 21-year-old having scored 61 goals in 66 league games for the Ruhr valley club since joining them in early 2020.
Several top clubs including Real Madrid, Paris St Germain and City, among others, have been linked with the player at times.
Sky Germany, which holds the rights to the Bundesliga, reported on Monday the player had informed the club of his wish to leave and that Manchester City had contacted the club over the player's release clause.
Several other media, including Bild newspaper and Kicker magazine, citing unnamed sources, also said a deal could be finalised in the coming days.
The club did not comment on the matter on Monday.
Dortmund are interested in Salzburg forward Karim Adeyemi as a possible Haaland replacement.
Dortmund sports director Sebastian Kehl had said on Sunday that the club wanted to have clarity on the matter of Haaland this week.