The 21-year-old, who has won three caps for Germany, is under contract at Hamburg until June 30, 2012, but the Hamburger Abendblatt newspaper report the transfer to Manchester is set to go through.
Boateng's advisor Jörg Neubauer told German agency SID, a subsidiary of AFP, that a decision is expected to be made soon.
If the centre-back joins City, he will be the third Hamburg player to switch to Manchester in the last two years.
In August 2008, Belgian international Vincent Kompany transferred to City for €8.5 million. Five months later Dutch midfielder Nigel de Jong joined him in the north of England for €18 million.