The reported Tottenham Hotspur target, who scored 24 league goals last season as surprise packages Genoa bagged a Europa League spot, has penned a four-year deal with the Italian champions.
An Inter statement gave no details of the fee.
Genoa team mate Thiago Motta, the Brazilian midfielder whose career was threatened by injury when at Barcelona and Atletico Madrid, is also due to be unveiled as an Inter player in the next few days.
Italy Under-21 striker Robert Acquafresca, on loan at Cagliari last season, has moved to Genoa from Inter as part of the deal but is being immediately sent to Atalanta.
Genoa are also in the process of signing Inter's Argentine forward Hernan Crespo and Atalanta striker Sergio Floccari.
Inter coach Jose Mourinho, hoping to cling on to restless striker Zlatan Ibrahimovic and right back Maicon, is keen to buy Ricardo Carvalho and Deco from former club Chelsea while Twente Enschede forward Marko Arnautovic is close to signing.
To make room for the new players which Mourinho hopes will give Inter more edge in the Champions League, forward Julio Cruz is leaving with Fiorentina expressing an interest.
Media reports say French midfielder Patrick Vieira wants to return to France.
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