Boateng started his career with Hertha Berlin before moving to Europa League semi-finalists Hamburg in 2007 and is the younger brother of Portsmouth midfielder Kevin-Prince Boateng.
The brothers could end up playing each other at the World Cup as the Pompey player has elected to represent Germany's Group D rivals Ghana.
And Mancini added: "It's good for Jerome to have agreed terms ahead of the World Cup, so he can now concentrate on playing for Germany.
"I will be very interested to see how he gets on in the tournament, before welcoming him back to meet his team-mates in pre-season."
Boateng follows current City players Vincent Kompany and Nigel de Jong in moving to Eastlands from Hamburg.
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